the function $f(x) = x + 7$ is one-to-one.

a. find an equation for $f^{-1}(x)$, the inverse function.
b. verify that your equation is correct by showing that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

a. select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answer. use integers or fractions for any numbers in the expression.)

a. $f^{-1}(x) = \quad$, for $x \le \quad$
b. $f^{-1}(x) = x - 7$, for all $x$
c. $f^{-1}(x) = \quad$, for $x \ge \quad$
d. $f^{-1}(x) = \quad$, for $x \
e \quad$

b. verify that the equation is correct.
$f(f^{-1}(x)) = f(\quad)$ and $f^{-1}(f(x)) = f^{-1}(\quad)$ substitute
$= \quad$ $= \quad$ simplify

Question

the function $f(x) = x + 7$ is one-to-one.

a. find an equation for $f^{-1}(x)$, the inverse function.
b. verify that your equation is correct by showing that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

a. select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answer. use integers or fractions for any numbers in the expression.)

a. $f^{-1}(x) = \quad$, for $x \le \quad$
b. $f^{-1}(x) = x - 7$, for all $x$
c. $f^{-1}(x) = \quad$, for $x \ge \quad$
d. $f^{-1}(x) = \quad$, for $x \
e \quad$

b. verify that the equation is correct.
$f(f^{-1}(x)) = f(\quad)$ and $f^{-1}(f(x)) = f^{-1}(\quad)$ substitute
$= \quad$ $= \quad$ simplify

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Inverse Functions", "Function Composition" ], "new_concepts": [], "current_concepts": [ "Inverse Functions", "Function Composition" ] } ## Find the inverse function $f^{-1}(x)$ $$ \begin{aligned} y &= x + 7 \ x &= y - 7 \ f^{-1}(x) &= x - 7 \quad \text{for all } x \end{aligned} $$ ## Verify the composition $f(f^{-1}(x))$ $$ \begin{aligned} f(f^{-1}(x)) &= f(x - 7) \ &= (x - 7) + 7 \ &= x \end{aligned} $$ ## Verify the composition $f^{-1}(f(x))$ $$ \begin{aligned} f^{-1}(f(x)) &= f^{-1}(x + 7) \ &= (x + 7) - 7 \ &= x \end{aligned} $$ ## Question a (A) $f^{-1}(x) = \Box$, for $x \le \Box$

(B) $f^{-1}(x) = x - 7$, for all $x$

(C) $f^{-1}(x) = \Box$, for $x \ge \Box$

(D) $f^{-1}(x) = \Box$, for $x e \Box$ ## Question b $$ \begin{aligned} f(f^{-1}(x)) &= f(x - 7) \ &= x \end{aligned} $$ and $$ \begin{aligned} f^{-1}(f(x)) &= f^{-1}(x + 7) \ &= x \end{aligned} $$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Inverse Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the inverse function \(f^{-1}(x)\)

Verify the composition \(f(f^{-1}(x))\)

Verify the composition \(f^{-1}(f(x))\)

Answer:

Question a

Question b