Math  /  Algebra

QuestionThe function f(x)=x311f(x)=x^{3}-11 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 xx \neq \square B. f1(x)=x+113f^{-1}(x)=\sqrt[3]{x+11}, for all xx C. f1(x)=f^{-1}(x)= \square , for xx \leq \square D. f1(x)=f^{-1}(x)= \square , for xx \geq \square b. Verify that the equation is correct. f(f1(x))=f() and f1(f(x))=f1()= Substitute. = Simplify. \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\square) & \text { and } & f^{-1}(f(x)) & =f^{-1}(\square) & \\ & =\square & & \text { Substitute. } \\ & =\square & & \text { Simplify. } \end{array}

Studdy Solution
Verify f1(f(x))=x f^{-1}(f(x)) = x .
Substitute f(x)=x311 f(x) = x^3 - 11 into f1(x) f^{-1}(x) :
f1(f(x))=f1(x311) f^{-1}(f(x)) = f^{-1}(x^3 - 11)
=(x311)+113 = \sqrt[3]{(x^3 - 11) + 11}
=x33 = \sqrt[3]{x^3}
=x = x
This confirms that f1(f(x))=x f^{-1}(f(x)) = x .
The inverse function is f1(x)=x+113 f^{-1}(x) = \sqrt[3]{x + 11} and it satisfies both verification conditions.

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