APPC U2.7 Compositions of Functions

Let f(x) = 2x + 3 and g(x) = x^2. Evaluate the composite function (f ∘ g)(2).

• 7

• 11

• 19

• 27

Answer explanation

To evaluate (f ∘ g)(2), first find g(2): g(2) = 2^2 = 4. Then, find f(g(2)): f(4) = 2(4) + 3 = 8 + 3 = 11. Thus, the correct answer is 11.

Tags

CCSS.HSF-BF.A.1C

Given f(x) = √x and g(x) = 3x + 1, which of the following represents (g ∘ f)(9)?

• √9

• 3√9 + 1

• √(3 × 9 + 1)

• 3(√x + 1)

Answer explanation

To find (g ∘ f)(9), first calculate f(9) = √9 = 3. Then, substitute this into g: g(3) = 3(3) + 1 = 9 + 1 = 10. The correct representation is 3√9 + 1, which simplifies to 10.

Tags

CCSS.HSF-BF.A.1C

If f(x) = 4x - 5 and g(x) = (x + 5)/4, what is (f ∘ g)(x)?

• x

• 4x - 5

• (4x - 5)/4

• 16x - 20

Answer explanation

To find (f ∘ g)(x), substitute g(x) into f: f(g(x)) = f((x + 5)/4) = 4((x + 5)/4) - 5 = x + 5 - 5 = x. Thus, the correct answer is x.

Tags

CCSS.HSF-BF.A.1C

Given f(x) = e^x and g(x) = ln(x), which of the following is true about the compositions (f ∘ g)(x) and (g ∘ f)(x)?

• Both compositions equal x

• (f ∘ g)(x) = x and (g ∘ f)(x) = e^x

• (f ∘ g)(x) = e^(ln(x)) and (g ∘ f)(x) = ln(e^x)

• Both compositions are different from x

Answer explanation

The composition (f ∘ g)(x) = f(g(x)) = f(ln(x)) = e^(ln(x)) = x. Similarly, (g ∘ f)(x) = g(f(x)) = g(e^x) = ln(e^x) = x. Thus, both compositions equal x.

Tags

CCSS.HSF-BF.A.1C

Evaluate (f ∘ g)(3) where f(x) = 5 - 2x and g(x) = x^2 + 1.

• 5

• -13

• -15

• -1

Answer explanation

First, calculate g(3): g(3) = 3^2 + 1 = 10. Then, evaluate f(g(3)): f(10) = 5 - 2(10) = 5 - 20 = -15. Thus, (f ∘ g)(3) = -15.

Tags

CCSS.HSF-BF.A.1C

Given the functions f(x) = 1/x and g(x) = x + 2, which of the following represents (f ∘ g)(x)?

• 1/x + 2

• 1/(x + 2)

• 1/x × 2

• 1/(2x)

Answer explanation

To find (f ∘ g)(x), we substitute g(x) into f(x). Thus, (f ∘ g)(x) = f(g(x)) = f(x + 2) = 1/(x + 2). Therefore, the correct answer is 1/(x + 2).

Tags

CCSS.HSF-BF.A.1C

Which of the following statements correctly describes the relationship between (f ∘ g)(x) and (g ∘ f)(x)?

• They are always equal

• They are never equal

• They are equal only if f and g are inverse functions

• They are equal only if f(x) = g(x

Answer explanation

(f ∘ g)(x) and (g ∘ f)(x) are generally not equal. They are equal only if f and g are inverse functions, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains.

Tags

CCSS.HSF-BF.A.1C

If \( h(x) = f(g(x)) \) and \( f(x) = 2x \), what is \( h(x) \) when \( g(x) = x^3 - x \)?

• \( 2x^3 - 2x \)

• \( (2x)^3 - 2x \)

• \( 2(x^3 - x) \)

• \( 2x^3 + 2x \)

Answer explanation

To find h(x), substitute g(x) into f: h(x) = f(g(x)) = f(x^3 - x) = 2(x^3 - x). Thus, the correct answer is 2(x^3 - x).

Tags

CCSS.HSF-BF.A.1C

Rewrite the function \( k(x) = 3x + 4 \) as a composition of two simpler functions.

• \( k(x) = f(g(x)) \) where \( f(x) = 3x \) and \( g(x) = x + 4 \)

• \( k(x) = f(g(x)) \) where \( f(x) = x + 4 \) and \( g(x) = 3x \)

• \( k(x) = f(g(x)) \) where \( f(x) = 3x + 4 \) and \( g(x) = x \)

• \( k(x) = f(g(x)) \) where \( f(x) = 3x \) and \( g(x) = x \)

Answer explanation

The correct choice is $ k(x) = f(g(x)) $ where $ f(x) = x + 4 $ and $ g(x) = 3x $. This composition first applies $ g(x) $ to multiply by 3, then $ f(x) $ adds 4, resulting in $ k(x) = 3x + 4 $.

Tags

CCSS.HSF-BF.A.1C

Consider the additive transformation \( f(x) = (x^2) + 5 \). Which composition represents this transformation?

• \( f(x) = g(h(x)) \) where \( g(x) = x + 5 \) and \( h(x) = x^2 \)

• \( f(x) = g(h(x)) \) where \( g(x) = x^2 \) and \( h(x) = x + 5 \)

• \( f(x) = g(x) + h(x) \) where \( g(x) = x^2 \) and \( h(x) = 5 \)

• \( f(x) = g(x) \times h(x) \) where \( g(x) = x^2 \) and \( h(x) = 5 \)

Answer explanation

The transformation f(x) = x^2 + 5 can be expressed as g(h(x)) where h(x) = x^2 and g(x) = x + 5. This means first squaring x and then adding 5, matching the original function.

Tags

CCSS.HSF.BF.B.3