Math  /  Algebra

QuestionThe function f(x)=x3+5f(x)=x^{3}+5 is one-to-one. a. Find an equation for f1f^{-1}, the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-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. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \geq \square B. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \leq \square C. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \neq \square D. f1(x)=x53f^{-1}(x)=\sqrt[3]{x-5}, for all xx b. Verify that the equation is correct. f(f1(x))=f() and f1(f(x))=f1:() Substitute. =\begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\square) & \text { and } & f^{-1}(f(x)) & =f^{-1}:(\square) & \\ & & & \text { Substitute. } \\ & =\square & & & & \end{array}

Studdy Solution
Verify that f1(f(x))=x f^{-1}(f(x)) = x :
Start with f1(f(x)) f^{-1}(f(x)) :
f1(f(x))=f1(x3+5) f^{-1}(f(x)) = f^{-1}(x^3 + 5)
Substitute into the inverse function:
=(x3+5)53 = \sqrt[3]{(x^3 + 5) - 5}
Simplify:
=x33 = \sqrt[3]{x^3}
=x = x
The correct choice is D: f1(x)=x53 f^{-1}(x) = \sqrt[3]{x - 5} , for all x x .
Verification shows that both f(f1(x))=x f(f^{-1}(x)) = x and f1(f(x))=x f^{-1}(f(x)) = x are true.

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