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Problem 32 Hard Difficulty

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)?

Answer

a) $g^{-1}(x)=\sqrt[3]{1-x^{3}}$
$g$ and $g^{-1}$ are equivalent
b) See explanation

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Video Transcript

here we have function g of X, and we want to find its inverse. So let's start by re naming it as why. And then we're going to switch X and y. So now we have X equals the cube root of one minus y cubed. Now we want to solve for y isolate. Why? So we que both sides And then we subtract one from both sides. And then we could multiply both sides by negative one to get the wife cubed by itself. And finally to get why by itself we can cue brute both sides. And now, instead of calling it why we can call it g inverse of X. So let's take a look at what we have. G inverse of X is the cube root of one minus X cubed G of X was the cube root of one minus X cubed. They're exactly the same. Now let's take a look at the graph to understand why. So grabbing a graphing calculator going toe y equals we type in the function y equals a cube root of one minus X cubed and by the way, to find the cube root function, you can go to your math menu. There, you see it's number four, and I'm also going to graph y equals X just so I can see that reflection line. Now I'm going to go to Zoom Square for a good viewing window, and we can see that the graph that Cuba A graph is actually symmetric about the line y equals ax already, so it's just a reflection on itself.