Combine Different Function Types - Algebra 1

A projectile's height (in meters) is modeled as a quadratic term plus a constant initial height. Let $f(t)=-5t^2+20t$ and $g(t)=2$ . Write the combined function $h(t)=(f+g)(t)$ .
Answer: $h(t)=-5t^2+20t+2$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! To find (f + g)(t) where f(t) = -5t² + 20t and g(t) = 2, we add the expressions: (f + g)(t) = -5t² + 20t + 2, which is already in quadratic form. This new function combines the quadratic projectile motion with the constant initial height. Choice A correctly combines the functions by adding the quadratic and constant, giving h(t) = -5t² + 20t + 2. Choice C confuses addition with multiplication: (f + g)(t) means add the outputs, NOT multiply. These are very different operations! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Combining different function types creates new behaviors: adding a constant to an exponential (like y = $2^x$ + 3) shifts the exponential curve vertically, adding a linear to a quadratic changes the parabola's shape, and so on. Each function type brings its characteristic behavior to the combination!
A company's monthly profit is modeled as revenue minus cost. Revenue is $R(x)=50x$ dollars and cost is $C(x)=12x+300$ dollars, where $x$ is the number of units sold. What is the profit function $P(x)=(R-C)(x)$ ?
Answer: $P(x)=38x-300$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. Subtracting functions works the same way: (f - g)(x) = f(x) - g(x), and this is especially useful in real-world models like profit = revenue - cost. If R(x) = 50x (revenue) and C(x) = 20x + 500 (cost), then P(x) = R(x) - C(x) = 50x - (20x + 500) = 30x - 500. Don't forget to distribute that negative sign! The context describes a company's profit using linear functions for revenue and cost, so combining them with subtraction: P(x) = R(x) - C(x) = 50x - (12x + 300) = 38x - 300. This models profit as revenue minus expenses, with the linear terms handling sales and fixed/variable costs. Choice A correctly combines the functions by subtracting and simplifying to 38x - 300. Choice C makes a sign error when subtracting: it computes 50x - (12x + 300) as 38x + 300, but distributing the negative gives -12x - 300, so 50x - 12x - 300 = 38x - 300. That negative sign changes everything! The recipe for combining functions: (1) Write out what (f - g)(x) means: f(x) - g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully (watch signs!), (4) Simplify by combining like terms. For subtraction, always use parentheses around the second function: f(x) - (g(x)) helps you remember to distribute the negative!
A model compares two quantities by division. Let $f(x)=x+2$ (linear) and $g(x)=x-3$ (linear). Find the domain of $(f/g)(x)=\dfrac{f(x)}{g(x)}$ .
Answer: $\{x\mid x\neq 3\}$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. The domain of a combined function is where BOTH original functions are defined: if f needs x ≥ 0 and g needs x ≠ 2, then (f + g) needs both conditions: x ≥ 0 AND x ≠ 2. For division (f/g), we also need g(x) ≠ 0. It's the intersection of all the requirements! Looking at the domain of each function: f(x) = x + 2 requires all real x (no restrictions), and g(x) = x - 3 requires all real x (no restrictions). For (f/g)(x), we need BOTH conditions to be met (all real) plus we need g(x) ≠ 0, which means solving x - 3 = 0 gives x ≠ 3. Combining all restrictions: domain is all real x except x = 3. Choice A correctly identifies the domain by excluding where the denominator is zero, giving {x | x ≠ 3}. Choice D gives the domain for f or g alone, but misses that for division we need to exclude g(x) = 0: without this restriction, the function would be undefined at x = 3! For finding domains of combinations: make a list of ALL restrictions: (1) domain restrictions from f, (2) domain restrictions from g, (3) for (f/g), also where g(x) = 0. The domain of the combination is where ALL of these are satisfied simultaneously—the intersection of all conditions!
Let $f(x)=\sqrt{x}$ (radical) and $g(x)=x+1$ (linear). Evaluate $(f\cdot g)(4)$ .
Answer: $10$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using multiplication, we write (f · g)(x) = f(x) · g(x), which means 'multiply the outputs': for any input x, evaluate both f(x) and g(x), then multiply those results. For example, if f(x) = 2x and g(x) = x + 1, then (f · g)(x) = 2x · (x + 1) = 2x² + 2x. The result is a new function that scales one by the other! To evaluate (f · g)(4), we first find f(4) = √4 = 2, then find g(4) = 4 + 1 = 5. Now we multiply: 2 · 5 = 10. Alternatively, we could first combine to get (f · g)(x) = √x · (x + 1), then substitute 4 directly: √4 · (4 + 1) = 2 · 5 = 10. Choice A correctly evaluates at the given point by multiplying the outputs, giving 10. Choice B might come from adding instead: √4 + (4 + 1) = 2 + 5 = 7, but then maybe miscalculating; the question asks for multiplication, not addition. Make sure you're performing the operation that's actually requested! To evaluate a combined function at a specific value, you have two methods: (Method 1) Evaluate each function separately at that value, then combine the results, OR (Method 2) First combine the functions into one expression, then substitute the value. Both work—choose whichever feels easier for the problem! Common pitfall: don't confuse (f · g)(x) with f(g(x)). The notation (f · g)(x) means 'multiply the functions,' while f(g(x)) means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
Let $f(x)=\dfrac{1}{x}$ and $g(x)=x^2+1$ . Find the domain of $(f+g)(x)$ .
Answer: $\{x\mid x\ne 0\}$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. The domain of a combined function is where BOTH original functions are defined: if f needs x ≥ 0 and g needs x ≠ 2, then (f + g) needs both conditions: x ≥ 0 AND x ≠ 2. For division (f/g), we also need g(x) ≠ 0. It's the intersection of all the requirements! Looking at the domain of each function: f(x) = 1/x requires x ≠ 0, and g(x) = x² + 1 requires all real x (no restrictions). For (f + g)(x), we need BOTH conditions to be met: x ≠ 0 AND all real x, so just x ≠ 0. Choice B correctly identifies the domain as {x | x ≠ 0}. Choice C gives the domain for g alone, but misses that the combined function needs to satisfy BOTH domains: we need x ≠ 0 from f AND all reals from g, so the domain is {x | x ≠ 0}, not all reals. For finding domains of combinations: make a list of ALL restrictions: (1) domain restrictions from f, (2) domain restrictions from g, (3) for (f/g), also where g(x) = 0. The domain of the combination is where ALL of these are satisfied simultaneously—the intersection of all conditions!
Let $f(x)=x^2-4x+1$ and $g(x)=3x-5$ . Find $(f+g)(x)$ .
Answer: $(f+g)(x)=x^2-x-4$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! To find (f + g)(x) where f(x) = x² - 4x + 1 and g(x) = 3x - 5, we add the expressions: (f + g)(x) = (x² - 4x + 1) + (3x - 5) = x² - 4x + 3x + 1 - 5. Simplifying: x² - x - 4, a new function that combines quadratic and linear behaviors into another quadratic. Choice B correctly combines the functions by adding and simplifying to x² - x - 4. Choice A has a sign error when adding: it computes x² - 4x + 1 + 3x - 5 as x² - 7x + 6, but -4x + 3x is -x, not -7x, and 1 - 5 is -4, not +6—double-check your like terms! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully (watch signs!), (4) Simplify by combining like terms. Common pitfall: don't confuse (f + g)(x) with f(g(x)). The notation (f + g)(x) means 'add the functions,' while f(g(x)) means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
Let $f(x)=\sqrt{x}$ and $g(x)=x-4$ . Find the domain of $(f/g)(x)=\dfrac{f(x)}{g(x)}$ .
Answer: $\{x\mid x\ge 0,\ x\ne 4\}$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. The domain of a combined function is where BOTH original functions are defined: if f needs x ≥ 0 and g needs x ≠ 2, then (f + g) needs both conditions: x ≥ 0 AND x ≠ 2. For division (f/g), we also need g(x) ≠ 0. It's the intersection of all the requirements! Looking at the domain of each function: f(x) = √x requires x ≥ 0, and g(x) = x - 4 requires all real x (no restrictions). For (f / g)(x), we need BOTH conditions: x ≥ 0, plus we need g(x) ≠ 0, which means x - 4 ≠ 0 so x ≠ 4. Combining all restrictions: domain is x ≥ 0 and x ≠ 4. Choice C correctly identifies the domain as {x | x ≥ 0, x ≠ 4}. Choice B forgets to include the square root's restriction: it only excludes x = 4, but we also need x ≥ 0 for √x to be defined in reals. For finding domains of combinations: make a list of ALL restrictions: (1) domain restrictions from f, (2) domain restrictions from g, (3) for (f/g), also where g(x) = 0. The domain of the combination is where ALL of these are satisfied simultaneously—the intersection of all conditions!
A lab models the temperature of a cooling sample by adding a constant room temperature to an exponential cooling term. Let $R(t)=72$ (room temperature in °F) and $C(t)=18(0.85)^t$ . Write the combined temperature function $T(t)=(R+C)(t)$ .
Answer: $T(t)=72+18(0.85)^t$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! The context describes a cooling model's temperature using a constant for room temperature and an exponential for the cooling term, so combining them with addition: T(t) = R(t) + C(t) = 72 + 18 $(0.85)^t$ , which models the overall temperature as room temp plus the decaying difference. This models the sample approaching room temperature over time, with the constant providing the baseline and the exponential handling the decay. Choice A correctly combines the functions by adding the constant and exponential terms, giving T(t) = 72 + 18 $(0.85)^t$ . Choice B multiplies instead of adding: the question asks for addition, not multiplication—make sure you're performing the operation that's actually requested! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Real-world combinations make intuitive sense: total temperature = room temp + cooling difference (add), matching how we naturally combine quantities!
A lab models the temperature of a cooling sample by adding a constant room temperature to an exponential cooling term. Let $R(t)=72$ (room temperature in °F) and $C(t)=18(0.85)^t$ . Write the combined temperature function $T(t)=(R+C)(t)$ .
Answer: $T(t)=72+18(0.85)^t$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! The context describes a cooling model's temperature using a constant for room temperature and an exponential for the cooling term, so combining them with addition: T(t) = R(t) + C(t) = 72 + 18 $(0.85)^t$ , which models the overall temperature as room temp plus the decaying difference. This models the sample approaching room temperature over time, with the constant providing the baseline and the exponential handling the decay. Choice A correctly combines the functions by adding the constant and exponential terms, giving T(t) = 72 + 18 $(0.85)^t$ . Choice B multiplies instead of adding: the question asks for addition, not multiplication—make sure you're performing the operation that's actually requested! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Real-world combinations make intuitive sense: total temperature = room temp + cooling difference (add), matching how we naturally combine quantities!
Let $f(x)=|x|$ (absolute value) and $g(x)=x-4$ (linear). Simplify $(f+g)(x)$ .
Answer: $(f+g)(x)=|x|+x-4$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write $(f + g)(x) = f(x) + g(x)$ , which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if $f(x) = 2x + 1$ and $g(x) = x^2$ , then $(f + g)(x) = (2x + 1) + x^2 = x^2 + 2x + 1$ . The result is a new function that combines both behaviors! To find $(f + g)(x)$ where $f(x) = |x|$ and $g(x) = x - 4$ , we add the expressions: $(f + g)(x) = |x| + (x - 4) = |x| + x - 4$ . Simplifying: it can't be combined further without piecewise definition, but this form shows the absolute value plus linear. This new function combines the V-shaped absolute value with the linear shift. Choice B correctly combines the functions by adding the absolute value and linear terms, giving $(f + g)(x) = |x| + x - 4$ . Choice C confuses addition with multiplication: $(f + g)(x)$ means add the outputs, NOT multiply. These are very different operations! The recipe for combining functions: (1) Write out what $(f + g)(x)$ means: $f(x) + g(x)$ , (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Common pitfall: don't confuse $(f + g)(x)$ with $f(g(x))$ . The notation $(f + g)(x)$ means 'add the functions,' while $f(g(x))$ means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
A company's profit is revenue minus cost. Revenue is $R(x)=40x$ and cost is $C(x)=12x+300$ , where $x$ is the number of units sold. What is the profit function $P(x)=(R-C)(x)$ ?
Answer: $P(x)=28x-300$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. Subtracting functions works the same way: (f - g)(x) = f(x) - g(x), and this is especially useful in real-world models like profit = revenue - cost. If R(x) = 50x (revenue) and C(x) = 20x + 500 (cost), then P(x) = R(x) - C(x) = 50x - (20x + 500) = 30x - 500. Don't forget to distribute that negative sign! To find (R - C)(x) where R(x) = 40x and C(x) = 12x + 300, we subtract the expressions: (R - C)(x) = 40x - (12x + 300) = 40x - 12x - 300 = 28x - 300. Simplifying: combine like terms to get a linear function modeling profit. This new function combines the linear revenue with the linear cost (including fixed cost) to show net gain. Choice C correctly combines the functions by subtracting and simplifying, giving P(x) = 28x - 300. Choice A makes a sign error when subtracting: it computes 40x - 12x + 300 = 28x + 300, but when subtracting C(x), we need to distribute the negative to every term: 40x - (12x + 300) = 40x - 12x - 300. That negative sign changes everything! The recipe for combining functions: (1) Write out what (f - g)(x) means: f(x) - g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully (watch signs!), (4) Simplify by combining like terms. For subtraction, always use parentheses around the second function: f(x) - (g(x)) helps you remember to distribute the negative!
A model compares two quantities by division. Let $f(x)=x+2$ (linear) and $g(x)=x-3$ (linear). Find the domain of $(f/g)(x)=\dfrac{f(x)}{g(x)}$ .
Answer: $\{x\mid x\neq 3\}$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. The domain of a combined function is where BOTH original functions are defined: if f needs x ≥ 0 and g needs x ≠ 2, then (f + g) needs both conditions: x ≥ 0 AND x ≠ 2. For division (f/g), we also need g(x) ≠ 0. It's the intersection of all the requirements! Looking at the domain of each function: f(x) = x + 2 requires all real x (no restrictions), and g(x) = x - 3 requires all real x (no restrictions). For (f/g)(x), we need BOTH conditions to be met (all real) plus we need g(x) ≠ 0, which means solving x - 3 = 0 gives x ≠ 3. Combining all restrictions: domain is all real x except x = 3. Choice A correctly identifies the domain by excluding where the denominator is zero, giving {x | x ≠ 3}. Choice D gives the domain for f or g alone, but misses that for division we need to exclude g(x) = 0: without this restriction, the function would be undefined at x = 3! For finding domains of combinations: make a list of ALL restrictions: (1) domain restrictions from f, (2) domain restrictions from g, (3) for (f/g), also where g(x) = 0. The domain of the combination is where ALL of these are satisfied simultaneously—the intersection of all conditions!
A lab models the temperature (in °C) of a cooling liquid as room temperature plus an exponential cooling term. Let $f(t)=22$ and $g(t)=15(0.8)^t$ , where $t$ is in minutes. Find $(f+g)(t)$ .
Answer: $(f+g)(t)=22+15(0.8)^t$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! To find (f + g)(t) where f(t) = 22 and g(t) = 15 $(0.8)^t$ , we add the expressions: (f + g)(t) = 22 + 15 $(0.8)^t$ , which combines the constant room temperature with the exponential cooling term. This new function models the total temperature by blending steady-state and decaying behaviors. Choice B correctly combines the functions by adding the constant and exponential terms, giving (f + g)(t) = 22 + 15 $(0.8)^t$ . Choice A attempts to multiply but uses the wrong operation: the question asks for addition, not multiplication. Make sure you're performing the operation that's actually requested! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Combining different function types creates new behaviors: adding a constant to an exponential (like y = $2^x$ + 3) shifts the exponential curve vertically, adding a linear to a quadratic changes the parabola's shape, and so on. Each function type brings its characteristic behavior to the combination!
Let $f(x)=\sqrt{x}$ (radical) and $g(x)=x+1$ (linear). Evaluate $(f\cdot g)(4)$ .
Answer: $10$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using multiplication, we write (f · g)(x) = f(x) · g(x), which means 'multiply the outputs': for any input x, evaluate both f(x) and g(x), then multiply those results. For example, if f(x) = 2x and g(x) = x + 1, then (f · g)(x) = 2x · (x + 1) = 2x² + 2x. The result is a new function that scales one by the other! To evaluate (f · g)(4), we first find f(4) = √4 = 2, then find g(4) = 4 + 1 = 5. Now we multiply: 2 · 5 = 10. Alternatively, we could first combine to get (f · g)(x) = √x · (x + 1), then substitute 4 directly: √4 · (4 + 1) = 2 · 5 = 10. Choice A correctly evaluates at the given point by multiplying the outputs, giving 10. Choice B might come from adding instead: √4 + (4 + 1) = 2 + 5 = 7, but then maybe miscalculating; the question asks for multiplication, not addition. Make sure you're performing the operation that's actually requested! To evaluate a combined function at a specific value, you have two methods: (Method 1) Evaluate each function separately at that value, then combine the results, OR (Method 2) First combine the functions into one expression, then substitute the value. Both work—choose whichever feels easier for the problem! Common pitfall: don't confuse (f · g)(x) with f(g(x)). The notation (f · g)(x) means 'multiply the functions,' while f(g(x)) means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
Let $f(x)=x^2+2$ and $g(x)=x-1$ . Find $(f\cdot g)(x)$ .
Answer: $(f\cdot g)(x)=x^3-x^2+2x-2$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. Combining different function types creates new behaviors: adding a constant to an exponential (like y = $2^x$ + 3) shifts the exponential curve vertically, adding a linear to a quadratic changes the parabola's shape, and so on. Each function type brings its characteristic behavior to the combination! To find (f · g)(x) where f(x) = x² + 2 and g(x) = x - 1, we multiply the expressions: (f · g)(x) = (x² + 2)(x - 1) = x²(x - 1) + 2(x - 1) = x³ - x² + 2x - 2. Simplifying: x³ - x² + 2x - 2, a new cubic function combining quadratic and linear. Choice A correctly combines the functions by multiplying and expanding to x³ - x² + 2x - 2. Choice B has a sign error when multiplying: in (x² + 2)(x - 1), the x² * (-1) = -x², not +x²—watch the distribution! The recipe for combining functions: (1) Write out what (f · g)(x) means: f(x) · g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Common pitfall: don't confuse (f · g)(x) with f(g(x)). The notation (f · g)(x) means 'multiply the functions,' while f(g(x)) means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
Let $f(x)=\sqrt{x}$ and $g(x)=x-4$ . Find the domain of $(f/g)(x)=\dfrac{f(x)}{g(x)}$ .
Answer: $\{x\mid x\ge 0,\ x\ne 4\}$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. The domain of a combined function is where BOTH original functions are defined: if f needs x ≥ 0 and g needs x ≠ 2, then (f + g) needs both conditions: x ≥ 0 AND x ≠ 2. For division (f/g), we also need g(x) ≠ 0. It's the intersection of all the requirements! Looking at the domain of each function: f(x) = √x requires x ≥ 0, and g(x) = x - 4 requires all real x (no restrictions). For (f / g)(x), we need BOTH conditions: x ≥ 0, plus we need g(x) ≠ 0, which means x - 4 ≠ 0 so x ≠ 4. Combining all restrictions: domain is x ≥ 0 and x ≠ 4. Choice C correctly identifies the domain as {x | x ≥ 0, x ≠ 4}. Choice B forgets to include the square root's restriction: it only excludes x = 4, but we also need x ≥ 0 for √x to be defined in reals. For finding domains of combinations: make a list of ALL restrictions: (1) domain restrictions from f, (2) domain restrictions from g, (3) for (f/g), also where g(x) = 0. The domain of the combination is where ALL of these are satisfied simultaneously—the intersection of all conditions!
A company's monthly profit is modeled as revenue minus cost. Revenue is $R(x)=50x$ dollars and cost is $C(x)=12x+300$ dollars, where $x$ is the number of units sold. What is the profit function $P(x)=(R-C)(x)$ ?
Answer: $P(x)=38x-300$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. Subtracting functions works the same way: (f - g)(x) = f(x) - g(x), and this is especially useful in real-world models like profit = revenue - cost. If R(x) = 50x (revenue) and C(x) = 20x + 500 (cost), then P(x) = R(x) - C(x) = 50x - (20x + 500) = 30x - 500. Don't forget to distribute that negative sign! The context describes a company's profit using linear functions for revenue and cost, so combining them with subtraction: P(x) = R(x) - C(x) = 50x - (12x + 300) = 38x - 300. This models profit as revenue minus expenses, with the linear terms handling sales and fixed/variable costs. Choice A correctly combines the functions by subtracting and simplifying to 38x - 300. Choice C makes a sign error when subtracting: it computes 50x - (12x + 300) as 38x + 300, but distributing the negative gives -12x - 300, so 50x - 12x - 300 = 38x - 300. That negative sign changes everything! The recipe for combining functions: (1) Write out what (f - g)(x) means: f(x) - g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully (watch signs!), (4) Simplify by combining like terms. For subtraction, always use parentheses around the second function: f(x) - (g(x)) helps you remember to distribute the negative!
Let $h(x)=5$ and $k(x)=2^x$ . Find $(h\cdot k)(x)$ .
Answer: $(h\cdot k)(x)=5\cdot 2^x$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. Combining different function types creates new behaviors: adding a constant to an exponential (like y = $2^x$ + 3) shifts the exponential curve vertically, adding a linear to a quadratic changes the parabola's shape, and so on. Each function type brings its characteristic behavior to the combination! To find (h · k)(x) where h(x) = 5 and k(x) = $2^x$ , we multiply the expressions: (h · k)(x) = 5 · $2^x$ . Simplifying: already simple, a new function that combines constant and exponential behaviors into a scaled exponential. Choice B correctly combines the functions by multiplying to get 5 · $2^x$ . Choice C confuses multiplication with addition: it does 5 + $2^x$ , but the question asks for · (multiplication), not +. Make sure you're performing the operation that's actually requested! The recipe for combining functions: (1) Write out what (f · g)(x) means: f(x) · g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Common pitfall: don't confuse (f · g)(x) with f(g(x)). The notation (f · g)(x) means 'multiply the functions,' while f(g(x)) means 'composition' (plug g into f). If you see +, -, ×, or ÷ in the notation, you're combining outputs, not composing functions!
A projectile's height (in meters) is modeled as a quadratic term plus a constant initial height. Let $f(t)=-5t^2+20t$ and $g(t)=2$ . Write the combined function $h(t)=(f+g)(t)$ .
Answer: $h(t)=-5t^2+20t+2$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using addition, we write (f + g)(x) = f(x) + g(x), which means 'add the outputs': for any input x, evaluate both f(x) and g(x), then add those results. For example, if f(x) = 2x + 1 and g(x) = x², then (f + g)(x) = (2x + 1) + x² = x² + 2x + 1. The result is a new function that combines both behaviors! To find (f + g)(t) where f(t) = -5t² + 20t and g(t) = 2, we add the expressions: (f + g)(t) = -5t² + 20t + 2, which is already in quadratic form. This new function combines the quadratic projectile motion with the constant initial height. Choice A correctly combines the functions by adding the quadratic and constant, giving h(t) = -5t² + 20t + 2. Choice C confuses addition with multiplication: (f + g)(t) means add the outputs, NOT multiply. These are very different operations! The recipe for combining functions: (1) Write out what (f + g)(x) means: f(x) + g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Combining different function types creates new behaviors: adding a constant to an exponential (like y = $2^x$ + 3) shifts the exponential curve vertically, adding a linear to a quadratic changes the parabola's shape, and so on. Each function type brings its characteristic behavior to the combination!
A material's stretch factor is modeled by multiplying a constant calibration factor and an exponential response. Let $f(x)=3$ and $g(x)=2^x$ . Find $(f\cdot g)(x)$ .
Answer: $(f\cdot g)(x)=3\cdot 2^x$ This question tests your ability to combine standard function types—like linear, quadratic, exponential, and constant functions—using arithmetic operations to build more complex functions. When we combine functions using multiplication, we write (f · g)(x) = f(x) · g(x), which means 'multiply the outputs': for any input x, evaluate both f(x) and g(x), then multiply those results. For example, if f(x) = 2x and g(x) = x + 1, then (f · g)(x) = 2x · (x + 1) = 2x² + 2x. The result is a new function that scales one by the other! To find (f · g)(x) where f(x) = 3 and g(x) = $2^x$ , we multiply the expressions: (f · g)(x) = 3 · $2^x$ , which is already simplified as a scaled exponential function. This new function combines the constant calibration with the exponential response to model the stretch factor. Choice B correctly combines the functions by multiplying the constant and exponential, giving (f · g)(x) = 3 · $2^x$ . Choice D confuses multiplication with addition: the question asks for product, not sum. Make sure you're performing the operation that's actually requested! The recipe for combining functions: (1) Write out what (f · g)(x) means: f(x) · g(x), (2) Substitute the actual function expressions, (3) Perform the operation carefully, (4) Simplify by combining like terms. Real-world combinations make intuitive sense: stretch factor = calibration × response (multiply), which matches how we naturally combine quantities!

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