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!

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. In the context, w(d) gives the number of widgets after d days, and C(w) gives the total cost for w widgets. The composition C(w(d)) means first find widgets after d days, then find the cost for that many widgets. This represents the total cost after d days of production. For example, if w converts days to widgets and C converts widgets to dollars, then C(w(d)) converts days to dollars in one combined operation! Choice C correctly describes the composition by identifying it as the total cost after d days. Choice A reverses the order, describing something like the inverse or a different composition. 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! 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. f(g(x)) = nested = composition. f(x) + g(x) = side by side = addition. The notation tells you which operation to use!

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).

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.'

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.

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.

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 g(f(x)) tells you the order: work from the inside out, with f as the inner function and g as the outer function. If f(x) = 4 - x and g(x) = x², then g(f(x)) means 'plug 4 - x into g,' giving g(4 - x) = (4 - x)². You're replacing every x in g with the entire expression for f(x)! To find g(f(x)) where g(x) = x² and f(x) = 4 - x, we substitute the entire expression for f(x) into g: g(f(x)) = g(4 - x) = (4 - x)². Every place you see x in g, you replace it with 4 - x—the whole thing, treated as a single unit! Choice B correctly composes the functions by properly substituting f into g, giving (4 - x)². Choice A shows 16 - x², which might come from incorrectly thinking (4 - x)² = 4² - x² = 16 - x². Remember: (a - b)² ≠ a² - b²! We need to use the formula (a - b)² = a² - 2ab + b². To compose functions into a formula: (1) Write down the inner function's expression f(x) = 4 - x, (2) In the outer function g(x) = x², everywhere you see x, replace it with (4 - x)—use parentheses to treat it as one chunk!, (3) The result is (4 - x)², which equals 16 - 8x + x² when expanded.

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!

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(0)): Step 1 - find g(0) by substituting 0 into g; Step 2 - take that answer and substitute it into f. If g(0) = -3, then you need f(-3). It's like a relay race where g passes its output to f! To find f(g(0)), we work inside-out: First, g(0) = 0 - 3 = -3. Now we take this result and plug it into f: f(-3) = (-3)² + 2 = 9 + 2 = 11. So f(g(0)) = 11. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 11. Choice B only evaluates the inner function g(0) = -3 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(0) = -3. Step 2: Find f(-3) = 11. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier! Remember that when squaring a negative number like (-3)², the result is positive: (-3)² = 9, not -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(h(x))) can look intimidating, but it's just telling you the order: work from the inside out, just like nested parentheses in arithmetic—first h(x), then g of that, then f of that result—you're replacing step by step! To find f(g(h(x))) where f(x) = 2x + 3, g(x) = x², and h(x) = x - 1, we substitute inward: First, h(x) = x - 1; then g(h(x)) = (x - 1)²; then f(g(h(x))) = 2((x - 1)²) + 3 = 2(x - 1)² + 3—every place you see x in the outer, you build the chain! Choice B correctly composes the functions by properly substituting h into g into f, giving 2(x - 1)² + 3. Choice A makes a substitution error: when replacing with (x - 1) in g(x) = x², it's (x - 1)², not x² - 1, and then applying f correctly—we need to substitute the ENTIRE expression, including handling it as a grouped quantity! The golden rule for composition f(g(h(x))): work INSIDE OUT—the function closest to x (h) gets evaluated first, then g, then f—think: 'h hands to g, g hands to f.' To compose multiple functions: start with the innermost, substitute into the next, and so on, using parentheses to keep track!