Question
Let $g(x)=\sqrt[3]{1-x^3}$.(a) Find $g^{-1}$. How is it related to $g$ ?(b) Graph $g$. How do you explain your answer to part (a)?
Step 1
This is a cube root function where the expression inside the cube root is \( 1-x^3 \). Show more…
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(a) Use a graphing utility to graph both $g(x)=\sqrt[3]{2-x^{3}}$ and $y=x$ on the same set of axes. (b) What kind of symmetry do you observe? What does this tell you about the inverse for $g ?$ [If you are uncertain how to answer this last question, try coming back to it after working part (c). $]$ (c) Find the domain and range of $g$ and $g^{-1}$ then use algebra to find an explicit formula for $g^{-1}(x) .$ What do you observe?
Functions
Inverse Functions
(a) Use a graphing utility to graph both $g(x)=\sqrt[3]{2-x^{3}}$ and $y=x$ on the same set of axes. (b) What kind of symmetry do you observe? What does this tell you about the inverse for $g ?$ [If you are uncertain how to answer this last question, try coming back to it after working part (c).] (c) Starting with the equation $g(x)=\sqrt[3]{2-x^{3}}$, follow the two-step procedure on page 198 to find $g^{-1}(x) .$ What do you observe?
Show that $f$ and $g$ are inverse functions (a) algebraically and(b) graphically. $$ f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x} $$
Functions and Their Graphs
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