find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.

$f(x) = \sqrt3{x - 5}$ and $g(x) = x^3 + 5$

$f(g(x)) = \square$

$g(f(x)) = \square$

determine whether $f$ and $g$ are inverses of each other. choose the correct answer below.

a. functions $f$ and $g$ are inverses of each other.
b. functions $f$ and $g$ are not inverses of each other.

Question

find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other.

$f(x) = \sqrt3{x - 5}$ and $g(x) = x^3 + 5$

$f(g(x)) = \square$

$g(f(x)) = \square$

determine whether $f$ and $g$ are inverses of each other. choose the correct answer below.

a. functions $f$ and $g$ are inverses of each other.
b. functions $f$ and $g$ are not inverses of each other.

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Function Composition", "Inverse Functions" ], "new_concepts": [], "current_concepts": [ "Function Composition", "Inverse Functions" ] } ## Evaluate the composition f(g(x)) Using the *Function Composition* knowledge point $$ \begin{aligned} f(g(x)) &= f(x^3 + 5) \ &= \sqrt[3]{(x^3 + 5) - 5} \ &= \sqrt[3]{x^3} \ &= x \end{aligned} $$ ## Evaluate the composition g(f(x)) Using the *Function Composition* knowledge point $$ \begin{aligned} g(f(x)) &= g(\sqrt[3]{x - 5}) \ &= (\sqrt[3]{x - 5})^3 + 5 \ &= (x - 5) + 5 \ &= x \end{aligned} $$ ## Determine if the functions are inverses Using the *Inverse Functions* knowledge point $$ \begin{aligned} &\text{Since } f(g(x)) = x \text{ and } g(f(x)) = x,\ &\text{the functions } f \text{ and } g \text{ are inverses of each other.} \end{aligned} $$ ## Question 1 $f(g(x)) =$ $x$ ## Question 2 $g(f(x)) =$ $x$ ## Question 3 A. Functions f and g are inverses of each other.

B. Functions f and g are not inverses of each other.

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Inverse Functions" ] }

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Question

Explanation:

Evaluate the composition f(g(x))

Evaluate the composition g(f(x))

Determine if the functions are inverses

Answer:

Question 1

Question 2

Question 3