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Algebra Help: Compose Two Functions

Review real example questions for Compose Two Functions in Algebra.

Question 1 / 10

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A factory's total cost C(w)C(w) (in dollars) depends on the number of widgets ww produced: C(w)=50+8wC(w)=50+8w. The number of widgets produced after dd days is w(d)=12dw(d)=12d. What does the composition C(w(d))C(w(d)) represent?

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Question 1

A factory's total cost C(w)C(w) (in dollars) depends on the number of widgets ww produced: C(w)=50+8wC(w)=50+8w. The number of widgets produced after dd days is w(d)=12dw(d)=12d. What does the composition C(w(d))C(w(d)) represent?

  1. The number of days needed to produce ww widgets
  2. The total cost (in dollars) after dd days of production (correct answer)
  3. The number of widgets produced per dollar
  4. The cost per widget after dd days

Explanation: 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!

Question 2

Use the table to evaluate f(g(2))f(g(2)).

Table for gg:

  • g(2)=5g(2)=5

Table for ff:

  • f(5)=12f(5)=12​
  1. 55
  2. 1212 (correct answer)
  3. f(2)f(2)
  4. g(12)g(12)

Explanation: 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.

Question 3

A factory's cost (in dollars) to produce ww widgets is C(w)=5w+100C(w)=5w+100. The number of widgets produced after dd days is w(d)=20dw(d)=20d. What does C(w(d))C(w(d)) represent?

  1. The number of widgets produced after dd days
  2. The cost to produce the number of widgets made after dd days (correct answer)
  3. The cost to produce dd widgets
  4. The number of days needed to spend CC dollars

Explanation: This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. In the context, we have two functions: w(d) gives the number of widgets produced after d days, and C(w) gives the cost to produce w widgets. The composition C(w(d)) means we first find how many widgets are produced after d days, then find the cost to produce that many widgets. This represents a two-step process: days → widgets → cost. 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 describes the composition: 'The cost to produce the number of widgets made after d days.' This captures the two-step nature: first we determine the widget count from the days (using w), then we find the cost for that widget count (using C). Choice B reverses the interpretation, suggesting 'The cost to produce d widgets,' which would be C(d), not 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 composition represents chained processes: think of real-world sequences. If one function converts input A to output B, and another converts input B to output C, then their composition converts A directly to C. Here, days → widgets → cost, so C(w(d)) takes days and gives cost!

Question 4

If f(x)=2x−5f(x)=2x-5 and g(x)=x2+1g(x)=x^2+1, find f(g(x))f(g(x)).

  1. 2x2−32x^2-3 (correct answer)
  2. (2x−5)(x2+1)(2x-5)(x^2+1)
  3. x2+1x^2+1
  4. 2x2−52x^2-5

Explanation: 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) = 2x - 5 and g(x) = x² + 1, then f(g(x)) means 'plug x² + 1 into f,' giving f(x² + 1) = 2(x² + 1) - 5 = 2x² + 2 - 5 = 2x² - 3. To find f(g(x)) where f(x) = 2x - 5 and g(x) = x² + 1, we substitute the entire expression for g(x) into f: f(g(x)) = f(x² + 1) = 2(x² + 1) - 5 = 2x² + 2 - 5 = 2x² - 3—every place you see x in f, you replace it with (x² + 1)—the whole thing, not just x! Choice C correctly composes the functions by properly substituting g into f, giving 2x² - 3. Choice A has the right process but makes an arithmetic error by not distributing the 2 to the +1, resulting in 2x² - 5 instead of 2x² - 3—when composing functions, there are multiple steps where calculation errors can creep in—always double-check each substitution and simplification! To compose functions into a formula: (1) Write down the inner function's expression g(x) = x² + 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.

Question 5

If f(x)=x+4f(x)=x+4 and g(x)=2x2g(x)=2x^2, find both f(g(x))f(g(x)) and g(f(x))g(f(x)).

  1. f(g(x))=2(x+4)2f(g(x))=2(x+4)^2 and g(f(x))=2x2+4g(f(x))=2x^2+4
  2. f(g(x))=2x2+4f(g(x))=2x^2+4 and g(f(x))=2x2+4g(f(x))=2x^2+4
  3. f(g(x))=2x2+4f(g(x))=2x^2+4 and g(f(x))=2(x+4)2g(f(x))=2(x+4)^2 (correct answer)
  4. f(g(x))=2x2f(g(x))=2x^2 and g(f(x))=x+4g(f(x))=x+4

Explanation: 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!

Question 6

If f(x)=2x+3f(x)=2x+3, g(x)=x2g(x)=x^2, and h(x)=x−1h(x)=x-1, find f(g(h(x)))f(g(h(x))).

  1. 2(x2−1)+32(x^2-1)+3
  2. 2(x−1)2+32(x-1)^2+3 (correct answer)
  3. 2x2+32x^2+3
  4. (2x+3)2−1(2x+3)^2-1

Explanation: 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!

Question 7

If f(x)=2x+1f(x)=2x+1 and g(x)=x−3g(x)=x-3, which expression equals f(g(x))f(g(x))?

  1. 2x−52x-5 (correct answer)
  2. 2x−22x-2
  3. x−2x-2
  4. 2x+1−32x+1-3

Explanation: 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. To find f(g(x)) where f(x) = 2x + 1 and g(x) = x - 3, we substitute the entire expression for g(x) into f: f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5. Every place you see x in f, you replace it with x - 3—the whole thing, not just x! Choice A correctly composes the functions by properly substituting g into f, giving 2x - 5. Choice B has an arithmetic error in the simplification (should be -6 + 1 = -5, not -2), Choice C seems to only use part of f(x), and Choice D shows the unsimplified form but equals 2x - 2, not 2x - 5. The golden rule for composition f(g(x)): work INSIDE OUT, just like nested parentheses. To compose: (1) Write g(x) = x - 3, (2) In f(x) = 2x + 1, replace every x with (x - 3), (3) Simplify: 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5.

Question 8

If f(x)=4xf(x)=4x and g(x)=x2−3g(x)=x^2-3, what is f(g(x))f(g(x))?​

  1. 4x2−34x^2-3
  2. (4x)2−3(4x)^2-3
  3. 4(x2−3)4(x^2-3) (correct answer)
  4. 4x(x2−3)4x(x^2-3)

Explanation: 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) = 4x and g(x) = x² - 3, then f(g(x)) means 'plug x² - 3 into f,' giving f(x² - 3) = 4(x² - 3). You're replacing every x in f with the entire expression for g(x)! To find f(g(x)) where f(x) = 4x and g(x) = x² - 3, we substitute the entire expression for g(x) into f: f(g(x)) = f(x² - 3) = 4(x² - 3) = 4x² - 12. Every place you see x in f, you replace it with (x² - 3)—the whole thing, not just x! Choice C correctly composes the functions by properly substituting g into f, giving 4(x² - 3). Choice A makes a substitution error: when replacing x with (x² - 3) in f(x) = 4x, we need to substitute the ENTIRE expression for g(x), including handling it as a grouped quantity. We get 4(x² - 3), not 4x² - 3! To compose functions into a formula: (1) Write down the inner function's expression g(x) = x² - 3, (2) In the outer function f(x) = 4x, everywhere you see x, replace it with (x² - 3)—treat it as a single chunk, use parentheses!, (3) Keep the result as 4(x² - 3) or expand to 4x² - 12 if needed. The golden rule for composition f(g(x)): work INSIDE OUT, and always use parentheses when substituting to show that you're multiplying 4 by the entire expression (x² - 3)!

Question 9

Suppose f(x)=3x+2f(x)=3x+2 and g(x)=x−5g(x)=x-5. Which expression represents f(g(x))f(g(x))?

  1. 3x+2+(x−5)3x+2+(x-5)
  2. (3x+2)−5(3x+2)-5
  3. 3x−5+23x-5+2
  4. 3(x−5)+23(x-5)+2 (correct answer)

Explanation: 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) = 3x + 2 and g(x) = x - 5, then f(g(x)) means 'plug x - 5 into f,' giving f(x - 5) = 3(x - 5) + 2. You're replacing every x in f with the entire expression for g(x)! To find f(g(x)) where f(x) = 3x + 2 and g(x) = x - 5, we substitute the entire expression for g(x) into f: f(g(x)) = f(x - 5) = 3(x - 5) + 2 = 3x - 15 + 2 = 3x - 13. Every place you see x in f, you replace it with (x - 5)—the whole thing, not just x! Choice B correctly represents the composition by properly substituting g into f, showing 3(x - 5) + 2. Choice A adds the functions (3x + 2) + (x - 5) instead of composing them. 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! 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!

Question 10

Given f(x)=2x−5f(x)=2x-5 and g(x)=x2+1g(x)=x^2+1, find f(g(x))f(g(x)).

  1. 2x2−32x^2-3 (correct answer)
  2. 2x−42x-4
  3. x2−9x^2-9
  4. 2x2−52x^2-5

Explanation: 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) = 2x - 5 and g(x) = x² + 1, we substitute the entire expression for g(x) into f: f(g(x)) = f(x² + 1) = 2(x² + 1) - 5 = 2x² + 2 - 5 = 2x² - 3. Every place you see x in f, you replace it with (x² + 1)—the whole thing, not just x! Choice A correctly composes the functions by properly substituting g into f, giving 2x² - 3. Choice D makes an arithmetic error: when expanding 2(x² + 1) - 5, we get 2x² + 2 - 5 = 2x² - 3, not 2x² - 5! When composing functions, there are multiple steps where calculation errors can creep in—always double-check each substitution and simplification! To compose functions into a formula: (1) Write down the inner function's expression g(x) = x² + 1, (2) In the outer function f(x) = 2x - 5, everywhere you see x, replace it with (x² + 1)—treat it as a single chunk, use parentheses!, (3) Simplify by expanding and combining like terms.