The functions in Exercises 11-28 are all one-to-one. For each fun...

Question

The functions in Exercises 11-28 are all one-to-one. For each function,a. Find an equation for f^-1(x), the inverse function.b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = √x

Accepted Answer

Hello. Today we're going to be finding the inverse of our given function and verifying that that inverse works. So the function given to us is F of X is equal to eight times the square root of three X. Now, in order to start finding the inverse, we need to go ahead and rewrite F of X as Y. Then we want to go ahead and swap the positions of X and Y. Doing so is going to give me X is equal to eight times the square root of three Y. And in order to find our inverse, we need to go ahead and solve for the variable Y. Now in order to sulfur Y, I want to go ahead and get the square root by itself. So I need to divide both sides of the equation by eight. Doing so is going to give me X over eight is equal to the square root of three Y. And now I need to go ahead and get rid of the square root. And in order to do that I can go ahead and square both sides of the equation. That's going to give me X over eight squared which is X squared over 64 equal to three Y. And in order to get Y by itself, I need to divide both sides of the equation by three. So what that leaves me with is X squared over 64. All over three equal to Y. So we have a complex fraction on the left hand side of this equation. Well I can go ahead and rewrite three as 3/1 and to get rid of the complex fraction, I can go ahead and multiply both the numerator and denominator by the reciprocal of the denominator. The reciprocal is going to be 1/3. That's going to cancel the complex denominator. And what I am left with is X squared over times three equal to y, 64 times three Is going to equal to 192. So why is equal to x squared over 192. So this is going to be our f inverse. But now we need to verify that that is our f inverse. In order to verify that we need to find two composite functions F of f inverse of X and F inverse of F of X. If these two functions are equal to each other and produce the value of X, then they are in verses of each other or at least that verifies that the inverse works. So let's go ahead and start by finding f f f inverse of X. What that means is we're going to take f inverse and plug it into our original equation and that's going to give us eight times the square root of three times x squared over 192. Now, three and 192 can reduce each other as three reduces to one and 100 92 reduces to 64. That leaves us with eight times the square root of x squared over 64. And that means we need to go ahead and take the square root of both the numerator and denominator. While the square root of X squared is just X. And the square root of 64 is going to leave us with eight. So we are left with eight times the quantity X over eight. And we can go ahead and simplify the eights in both the numerator and denominator and that's going to leave us with X. So that verifies the F of f inverse of X are in verses. But now we need to verify f inverse of F of X. So F inverse of F of X is going to be eight times the square root three X. That quantity squared all over 192. Now, eight squared is going to give us 64 the square root of three X squared is just going to cancel the square root. So we are left with 64 times the quantity three X over 192 64 times three X is going to give us 192 x over 192. And we can go ahead and cancel out the 192 in both the numerator and denominator. And that's going to leave us with just X. So f of f inverse of X is equal to f inverse of F of X and both are equal to X. So that verifies that are inverse function works. So that means that the inverse of F of X is going to be F inverse is equal to X squared over 192. And the answer to this problem is going to be D. So I hope this video helps you understand you how to find an inverse and verify it, and I'll go ahead and see you all in the next video.

The functions in Exercises 11-28 are all one-to-one. For each function,a. Find an equation for f^-1(x), the inverse function.b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = √x

Key Concepts

One-to-One Functions

Inverse Functions

Verification of Inverse Functions

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