Compose Two Functions - Algebra 1

1. For $f(x)=\sqrt{x}$ and $g(x)=x+15$, evaluate $f(g(1))$.​
   Answer: $4$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(1)): Step 1 - find g(1) by substituting 1 into g; Step 2 - take that answer and substitute it into f. If g(1) = 16, then you need f(16). It's like a relay race where g passes its output to f! To find f(g(1)), we work inside-out: First, g(1) = 1 + 15 = 16. Now we take this result and plug it into f: f(16) = √16 = 4. So f(g(1)) = 4. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 4. Choice D adds the functions (√1 + 15) instead of composing them (f(g(1))). These are completely different operations: addition of functions combines outputs, while composition plugs the output of one into the other as input. Watch out for this confusion! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = 16. Step 2: Find f(16) = 4. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier! Quick check to see if you have composition vs. addition: composition has one function INSIDE another (literally nested), while addition has functions side by side with a + between them.
2. Given $f(x)=\sqrt{x+4}$ and $g(x)=3x-1$, compute $(f\circ g)(5)$.
   Answer: $\sqrt{18}$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(3)): Step 1 - find g(3) by substituting 3 into g; Step 2 - take that answer and substitute it into f—if g(3) = 7, then you need f(7)—it's like a relay race where g passes its output to f! To find (f ∘ g)(5) where f(x) = √(x + 4) and g(x) = 3x - 1, we work inside-out: First, g(5) = 3(5) - 1 = 15 - 1 = 14; now we take this result and plug it into f: f(14) = √(14 + 4) = √18—so (f ∘ g)(5) = √18. Choice A correctly evaluates the composition by working inside-out, giving √18. Choice B only evaluates the inner function g(5) = 14 but forgets the second step of plugging that into f, or perhaps miscalculates √(14 + 4) as √(14 + 1)—you're not done until you've applied both functions in order! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = some number; Step 2: Find f(that number) = final answer—write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier! The golden rule for composition f(g(x)): work INSIDE OUT, just like nested parentheses—the function closest to x (g in this case) gets evaluated first, then its output becomes the input for the next function (f).
3. A company's total cost $C(w)$ depends on the number of widgets $w$ produced, and the number of widgets produced depends on days $d$ according to $w(d)$. What does the composition $C(w(d))$ represent?​
   Answer: The total cost as a function of days This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Function composition C(w(d)) means 'C of w of d': first evaluate w(d), then take that result and plug it into C. Think of it like a two-step process or a chain: d goes into w, w(d) comes out, that goes into C, and C(w(d)) comes out. The inner function (w) is evaluated first, then the outer function (C) is applied to that result. In the context, w(d) gives the number of widgets produced after d days and C(w) gives the total cost for producing w widgets. The composition C(w(d)) means we're finding the cost of the widgets produced after d days: first w(d) tells us how many widgets are produced after d days, then C takes that number and tells us the total cost. This represents the total cost as a function of days. For example, if w converts days to widgets and C converts widgets to cost, then C(w(d)) converts days to cost in one combined operation! Choice C correctly identifies that C(w(d)) represents the total cost as a function of days. Choice A reverses the order, giving w(C(d)) instead of C(w(d)). Remember: C(w(d)) means w is the inner function (evaluated first) and C is the outer function (applied second). The function name closest to d is evaluated first! To remember that order matters: think of real-world sequences. If you want to know the cost after 10 days, you first need to know how many widgets are made in 10 days (w(10)), then find the cost of that many widgets (C(w(10))). The composition C(w(d)) combines these two steps into one function that directly gives cost from days!
4. Given $f(x)=3x+2$ and $g(x)=x-4$, find both $f(g(x))$ and $g(f(x))$.​
   Answer: $f(g(x))=3x-10$ and $g(f(x))=3x-2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Order matters in composition: f(g(x)) is NOT the same as g(f(x)) in general! If f adds 2 after tripling and g subtracts 4, then f(g(x)) means 'subtract 4, then triple and add 2' (giving 3(x - 4) + 2 = 3x - 12 + 2 = 3x - 10), while g(f(x)) means 'triple and add 2, then subtract 4' (giving (3x + 2) - 4 = 3x - 2). Different order, different result! Let's find both directions: For f(g(x)), inner function g gives x - 4, and plugging into f gives f(x - 4) = 3(x - 4) + 2 = 3x - 12 + 2 = 3x - 10. For g(f(x)), inner function f gives 3x + 2, and plugging into g gives g(3x + 2) = (3x + 2) - 4 = 3x - 2. Notice how 3x - 10 ≠ 3x - 2—the order changes everything! Choice A correctly composes the functions in both orders by properly substituting and following the correct order, giving f(g(x)) = 3x - 10 and g(f(x)) = 3x - 2. Choice B reverses the order, giving g(f(x)) = 3x - 10 and f(g(x)) = 3x - 2. Remember: f(g(x)) means g is the inner function (evaluated first) and f is the outer function (applied second). The function name closest to x is evaluated first! To remember that order matters: think of real-world sequences. 'Put on socks then shoes' is different from 'put on shoes then socks'! Similarly, 'subtract 4 then triple and add 2' (giving 3x - 10) is different from 'triple and add 2 then subtract 4' (giving 3x - 2). In math, f(g(x)) and g(f(x)) are usually different, just like changing the order of getting dressed gives different results!
5. If $f(x)=x^2$ and $g(x)=2x-1$, what is $f(g(x))$?
   Answer: $(2x-1)^2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Function composition f(g(x)) means 'f of g of x': first evaluate g(x), then take that result and plug it into f. Think of it like a two-step process or a chain: x goes into g, g(x) comes out, that goes into f, and f(g(x)) comes out. The inner function (g) is evaluated first, then the outer function (f) is applied to that result. To find f(g(x)) where f(x) = x² and g(x) = 2x - 1, we substitute the entire expression for g(x) into f: f(g(x)) = f(2x - 1) = (2x - 1)². Every place you see x in f, you replace it with 2x - 1—the whole thing, not just x! Choice B correctly composes the functions by properly substituting g into f, giving (2x - 1)². Choice A makes a substitution error: when replacing x with 2x - 1 in f(x) = x², we need to substitute the ENTIRE expression for g(x), including handling it as a grouped quantity. We get (2x - 1)², not 4x² - 1! To compose functions into a formula: (1) Write down the inner function's expression g(x) = 2x - 1, (2) In the outer function f(x) = x², everywhere you see x, replace it with (2x - 1)—treat it as a single chunk, use parentheses!, (3) The result is (2x - 1)², which we could expand to 4x² - 4x + 1 if needed.
6. Let $f(x)=x-1$ and $g(x)=x^2+2x$. Compute $f(g(-1))$.
   Answer: $-2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(-1)): Step 1 - find g(-1) by substituting -1 into g; Step 2 - take that answer and substitute it into f. If g(-1) gives us some value, then we need f(that value). It's like a relay race where g passes its output to f! To find f(g(-1)) where f(x) = x - 1 and g(x) = x² + 2x, we work inside-out: First, g(-1) = (-1)² + 2(-1) = 1 + (-2) = 1 - 2 = -1. Now we take this result and plug it into f: f(-1) = -1 - 1 = -2. So f(g(-1)) = -2. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving -2. Choice B shows -1, which only completes the first step g(-1) = -1 but forgets the second step of plugging that into f. You're not done until you've applied both functions in order! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(-1) = (-1)² + 2(-1) = 1 - 2 = -1. Step 2: Find f(-1) = -1 - 1 = -2. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
7. If $f(x)=x^2-1$ and $g(x)=x+2$, what is $f(g(x))$?
   Answer: $ $(x+2)^2$ - 1 = $x^2$ + 4x + 3 $ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. The notation f(g(x)) can look intimidating, but it's just telling you the order: work from the inside out, just like nested parentheses in arithmetic. If $f(x) = x^2 - 1$ and $g(x) = x + 2$, then $f(g(x))$ means 'plug $x + 2$ into f,' giving $f(x + 2) = (x + 2)^2 - 1 = x^2 + 4x + 4 - 1 = x^2 + 4x + 3$. You're replacing every x in f with the entire expression for g(x)! To find $f(g(x))$ where $f(x) = x^2 - 1$ and $g(x) = x + 2$, we substitute the entire expression for g(x) into f: $f(g(x)) = f(x + 2) = (x + 2)^2 - 1 = x^2 + 4x + 4 - 1 = x^2 + 4x + 3$. Every place you see x in f, you replace it with $(x + 2)$—the whole thing, not just x! Choice B correctly composes the functions by properly substituting g into f, giving $(x + 2)^2 - 1 = x^2 + 4x + 3$. Choice C adds the functions ($f(x) + g(x)$) instead of composing them ($f(g(x)$)). These are completely different operations: addition of functions combines outputs, while composition plugs the output of one into the other as input. Watch out for this confusion! To compose functions into a formula: (1) Write down the inner function's expression $g(x) = whatever$ it is, (2) In the outer function $f(x)$, everywhere you see x, replace it with g(x) expression—treat it as a single chunk, use parentheses!, (3) Simplify by expanding and combining like terms. Example: $f(x) = x^2 - 1$, $g(x) = x + 2$, so $f(g(x)) = (x + 2)^2 - 1 = x^2 + 4x + 3$.
8. If $f(x)=x^2$ and $g(x)=x-1$, which expression represents $f(g(x))$?
   Answer: $(x-1)^2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. The notation f(g(x)) can look intimidating, but it's just telling you the order: work from the inside out, just like nested parentheses in arithmetic—if f(x) = x² and g(x) = x - 1, then f(g(x)) means 'plug x - 1 into f,' giving f(x - 1) = (x - 1)²—you're replacing every x in f with the entire expression for g(x)! To find f(g(x)) where f(x) = x² and g(x) = x - 1, we substitute the entire expression for g(x) into f: f(g(x)) = f(x - 1) = (x - 1)²—every place you see x in f, you replace it with (x - 1)—the whole thing, not just x! Choice B correctly composes the functions by properly substituting g into f, giving (x - 1)². Choice A multiplies the functions or subtracts incorrectly, giving x² - 1 instead of (x - 1)²—composition is NOT multiplication—it's substitution! f(g(x)) means substitute g(x) into f, not multiply f times g. To compose functions into a formula: (1) Write down the inner function's expression g(x) = x - 1, (2) In the outer function f(x), everywhere you see x, replace it with (g(x) expression)—treat it as a single chunk, use parentheses!, (3) Simplify by expanding and combining like terms—example: f(g(x)) = (x - 1)² = x² - 2x + 1, but the choice shows the compact form.
9. If $f(x)=x^2$ and $g(x)=2x-3$, what is $f(g(x))$?
   Answer: $(2x-3)^2=4x^2-12x+9$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. The notation f(g(x)) can look intimidating, but it's just telling you the order: work from the inside out, just like nested parentheses in arithmetic. If f(x) = x² and g(x) = 2x - 3, then f(g(x)) means 'plug 2x - 3 into f,' giving f(2x - 3) = (2x - 3)² = 4x² - 12x + 9. You're replacing every x in f with the entire expression for g(x)! To find f(g(x)) where f(x) = x² and g(x) = 2x - 3, we substitute the entire expression for g(x) into f: f(g(x)) = f(2x - 3) = (2x - 3)² = 4x² - 12x + 9. Every place you see x in f, you replace it with (2x - 3)—the whole thing, not just x! Choice B correctly composes the functions by properly substituting g into f, giving (2x - 3)² = 4x² - 12x + 9. Choice A makes a substitution error: when replacing x with 2x - 3 in f(x) = x², we need to substitute the ENTIRE expression, including handling it as a grouped quantity. For example, f(2x - 3) = (2x - 3)², not 2x² - 3! To compose functions into a formula: (1) Write down the inner function's expression g(x) = whatever it is, (2) In the outer function f(x), everywhere you see x, replace it with g(x) expression—treat it as a single chunk, use parentheses!, (3) Simplify by expanding and combining like terms. Example: f(x) = x², g(x) = 2x - 3, so f(g(x)) = (2x - 3)² = 4x² - 12x + 9.
10. Evaluate $(f\circ g)(2)$ for $f(x)=x^2$ and $g(x)=2x+1$.​​
    Answer: $25$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(2)): Step 1 - find g(2) by substituting 2 into g; Step 2 - take that answer and substitute it into f. If g(2) = 5, then you need f(5). It's like a relay race where g passes its output to f! To find f(g(2)), we work inside-out: First, g(2) = 2(2) + 1 = 4 + 1 = 5. Now we take this result and plug it into f: f(5) = 5² = 25. So f(g(2)) = 25. Two separate evaluations, one after the other! Choice B correctly evaluates the composition by working inside-out, giving 25. Choice A gives 9, which would be f(3), suggesting they might have made an arithmetic error when calculating g(2). For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = [some number]. Step 2: Find f([that number]) = [final answer]. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
11. Let $f(x)=x+6$ and $g(x)=5x$. Evaluate $f(g(2))$.
    Answer: $16$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(2)): Step 1 - find g(2) by substituting 2 into g; Step 2 - take that answer and substitute it into f. If g(2) gives some value, then you need f(that value). It's like a relay race where g passes its output to f! To find f(g(2)), we work inside-out: First, g(2) = 5(2) = 10. Now we take this result and plug it into f: f(10) = 10 + 6 = 16. So f(g(2)) = 16. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 16. Choice C might result from adding the functions instead of composing them, or from making an arithmetic error like computing g(2) = 5 + 2 = 7, then f(7) = 7 + 6 = 13, which doesn't match any choice. For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(2) = 10. Step 2: Find f(10) = 16. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
12. Let $f(x)=2x$ and $g(x)=x^2+1$. What is $f(g(3))$?
    Answer: $20$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(3)): Step 1 - find g(3) by substituting 3 into g; Step 2 - take that answer and substitute it into f. If g(3) = 10, then you need f(10). It's like a relay race where g passes its output to f! To find f(g(3)), we work inside-out: First, g(3) = 3² + 1 = 9 + 1 = 10. Now we take this result and plug it into f: f(10) = 2 * 10 = 20. So f(g(3)) = 20. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 20. Choice D has the right process but makes an arithmetic error by doing 2(3²) + 1 instead of 2(3² + 1), leading to 19. When composing functions, there are multiple steps where calculation errors can creep in—always double-check each substitution and simplification! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = some number. Step 2: Find f(that number) = final answer. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
13. Suppose $f(x)=\dfrac{x}{2}$ and $g(x)=x+8$. What is $(f\circ g)(x)$?
    Answer: $\dfrac{x+8}{2}$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. The notation (f∘g)(x) means f(g(x)): first evaluate g(x), then take that result and plug it into f. Think of it like a two-step process or a chain: x goes into g, g(x) comes out, that goes into f, and f(g(x)) comes out. The inner function (g) is evaluated first, then the outer function (f) is applied to that result. To find (f∘g)(x) where f(x) = x/2 and g(x) = x + 8, we substitute the entire expression for g(x) into f: (f∘g)(x) = f(g(x)) = f(x + 8) = (x + 8)/2. Every place you see x in f, you replace it with x + 8—the whole expression goes in the numerator! Choice B correctly composes the functions by properly substituting g into f, giving (x + 8)/2. Choice C shows x/2 + 4, which comes from incorrectly distributing the division: (x + 8)/2 ≠ x/2 + 8/2. When you divide a sum by 2, the entire sum stays in the numerator! The correct simplification is (x + 8)/2 = x/2 + 4, but the unsimplified form (x + 8)/2 is the direct result of composition. To compose functions into a formula: (1) Write down the inner function's expression g(x) = x + 8, (2) In the outer function f(x) = x/2, replace x with (x + 8), (3) This gives f(x + 8) = (x + 8)/2. The entire expression x + 8 becomes the new numerator!
14. Use the table to evaluate $f(g(2))$. Table for $g$: - $g(2)=5$ Table for $f$: - $f(5)=12$​
    Answer: $12$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(2)): Step 1 - find g(2) by looking it up in the table; Step 2 - take that answer and substitute it into f. If g(2) = 5, then you need f(5). It's like a relay race where g passes its output to f! To find f(g(2)), we work inside-out: First, from the table, g(2) = 5. Now we take this result and look up f(5) in the table: f(5) = 12. So f(g(2)) = 12. Two separate evaluations, one after the other! Choice B correctly evaluates the composition by working inside-out using the table values, giving 12. Choice A only evaluates the inner function g(2) = 5 but forgets the second step of plugging that into f. You're not done until you've applied both functions in order! For evaluating at a specific number using tables: do it in TWO separate steps: Step 1: Find g(input) = 5 from the table. Step 2: Find f(5) = 12 from the table. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier! When working with tables, the process is the same as with formulas: evaluate the inner function first, then use that result as the input for the outer function.
15. If $f(x)=x+4$ and $g(x)=2x^2$, find both $f(g(x))$ and $g(f(x))$.
    Answer: $f(g(x))=2x^2+4$ and $g(f(x))=2(x+4)^2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Order matters in composition: f(g(x)) is NOT the same as g(f(x)) in general! If f adds 4 and g squares and doubles, then f(g(x)) means 'square and double, then add 4', while g(f(x)) means 'add 4, then square and double'. Different order, different result! Let's find both directions: For f(g(x)), inner function g gives 2x², and plugging into f gives f(2x²) = 2x² + 4. For g(f(x)), inner function f gives x + 4, and plugging into g gives g(x + 4) = 2(x + 4)² = 2(x + 4)². Notice how 2x² + 4 ≠ 2(x + 4)²—the order changes everything! Choice B correctly composes the functions in both orders: f(g(x)) = 2x² + 4 and g(f(x)) = 2(x + 4)². Choice A incorrectly claims both compositions give the same result 2x² + 4. This would mean the functions commute, but they don't! When you square (x + 4), you get x² + 8x + 16, not just x² + 4. To remember that order matters: think of real-world sequences. 'Put on socks then shoes' is different from 'put on shoes then socks'! Similarly, 'square-and-double then add 4' (giving 2x² + 4) is different from 'add 4 then square-and-double' (giving 2(x + 4)²). In math, f(g(x)) and g(f(x)) are usually different, just like changing the order of getting dressed gives different results!
16. Given $f(x)=\sqrt{x}$ and $g(x)=x+7$, evaluate $f(g(9))$. Give your answer in simplest form.
    Answer: $4$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(9)): Step 1 - find g(9) by substituting 9 into g; Step 2 - take that answer and substitute it into f. If g(9) gives us some value, then we need f(that value). It's like a relay race where g passes its output to f! To find f(g(9)) where f(x) = √x and g(x) = x + 7, we work inside-out: First, g(9) = 9 + 7 = 16. Now we take this result and plug it into f: f(16) = √16 = 4. So f(g(9)) = 4. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 4. Choice B shows √16, which only completes the first step g(9) = 16 but forgets the second step of plugging that into f. You're not done until you've applied both functions in order! The golden rule for composition f(g(x)): work INSIDE OUT, just like nested parentheses. The function closest to x (g in this case) gets evaluated first, then its output becomes the input for the next function (f). For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(9) = 16. Step 2: Find f(16) = √16 = 4. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
17. A company's cost (in dollars) to make $w$ widgets is $C(w)=10w+50$. The number of widgets produced after $d$ days is $w(d)=20d$. What does $C(w(d))$ represent?
    Answer: The total cost to produce widgets after $d$ days This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Function composition f(g(x)) means 'f of g of x': first evaluate g(x), then take that result and plug it into f. In real-world contexts, this creates a chain of relationships: if one function converts A to B and another converts B to C, their composition converts A to C directly. In the context, w(d) = 20d tells us the number of widgets produced after d days, and C(w) = 10w + 50 tells us the cost to make w widgets. The composition C(w(d)) means: first find how many widgets are made in d days using w(d), then find the cost to make that many widgets using C. This represents the total cost to produce widgets after d days—we go from days to widgets to cost! Choice A correctly identifies that C(w(d)) represents the total cost to produce widgets after d days. Choice B reverses the relationship (that would be finding d from C), Choice C confuses the output (it gives cost, not widget count), and Choice D misinterprets the composition as division rather than substitution. To remember composition in context: think of it as a chain of conversions. Here, days → widgets → cost. The composition C(w(d)) takes you through the whole chain: input days, output cost. It's the cost after d days of production!
18. Given $f(x)=3x+2$ and $g(x)=x-4$, find both $f(g(x))$ and $g(f(x))$.
    Answer: $f(g(x))=3x-10,\; g(f(x))=3x-2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Order matters in composition: f(g(x)) is NOT the same as g(f(x)) in general! If f(x) = 3x + 2 and g(x) = x - 4, then f(g(x)) means 'apply g first, then f,' while g(f(x)) means 'apply f first, then g.' Different order, different result! Let's find both directions: For f(g(x)), inner function g gives x - 4, and plugging into f gives f(x - 4) = 3(x - 4) + 2 = 3x - 12 + 2 = 3x - 10. For g(f(x)), inner function f gives 3x + 2, and plugging into g gives g(3x + 2) = (3x + 2) - 4 = 3x - 2. Notice how 3x - 10 ≠ 3x - 2—the order changes everything! Choice A correctly computes both compositions: f(g(x)) = 3x - 10 and g(f(x)) = 3x - 2. Choice B reverses the answers, mixing up which composition gives which result. Remember: the function name closest to x is evaluated first! To remember that order matters: think of real-world sequences. 'Put on socks then shoes' is different from 'put on shoes then socks'! Similarly, 'subtract 4 then triple and add 2' gives a different result than 'triple and add 2 then subtract 4.'
19. Suppose $g(x)=x+4$ and $f(x)=5-x$. Compute $(f\circ g)(-1)$.
    Answer: $2$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(-1)): Step 1 - find g(-1) by substituting -1 into g; Step 2 - take that answer and substitute it into f. If g(-1) = 3, then you need f(3). It's like a relay race where g passes its output to f! To find f(g(-1)), we work inside-out: First, g(-1) = -1 + 4 = 3. Now we take this result and plug it into f: f(3) = 5 - 3 = 2. So f(g(-1)) = 2. Two separate evaluations, one after the other! Choice C correctly evaluates the composition by working inside-out, giving 2. Choice B has the right process but makes an arithmetic error in the second step: f(3) should be 5 - 3 = 2, not something leading to 10. When composing functions, there are multiple steps where calculation errors can creep in—always double-check each substitution and simplification! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = some number. Step 2: Find f(that number) = final answer. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!
20. Given $f(x)=x-5$ and $g(x)=4x$, find both $f(g(x))$ and $g(f(x))$.​​
    Answer: $f(g(x))=4x-5$ and $g(f(x))=4x-20$ This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. Order matters in composition: f(g(x)) is NOT the same as g(f(x)) in general! If f subtracts 5 and g multiplies by 4, then f(g(x)) means 'multiply by 4, then subtract 5' (giving 4x - 5), while g(f(x)) means 'subtract 5, then multiply by 4' (giving 4(x - 5) = 4x - 20). Different order, different result! Let's find both directions: For f(g(x)), inner function g gives 4x, and plugging into f gives f(4x) = 4x - 5. For g(f(x)), inner function f gives x - 5, and plugging into g gives g(x - 5) = 4(x - 5) = 4x - 20. Notice how 4x - 5 ≠ 4x - 20—the order changes everything! Choice A correctly composes the functions in both orders, giving f(g(x)) = 4x - 5 and g(f(x)) = 4x - 20. Choice B reverses the answers, mixing up which composition gives which result. Remember: f(g(x)) means g is the inner function (evaluated first) and f is the outer function (applied second). To remember that order matters: think of real-world sequences. 'Put on socks then shoes' is different from 'put on shoes then socks'! Similarly, 'multiply by 4 then subtract 5' (giving 4x - 5) is different from 'subtract 5 then multiply by 4' (giving 4x - 20).

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