The functions in Exercises 93–95 are all one-to-one. For each fun...

Question

The functions in Exercises 93–95 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(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = (x - 7)/(x + 2)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this function F of X equals x minus four. We want to figure out is this a 1 to 1 function? And if so, what is that inverse function? So let's quickly graph it to see if it passes a horizontal line test. We've got our horizontal X axis and our vertical Y axis, which is the same as F of X. And our function is F of X equals x minus four. And hey, that that's a line because if F of X is the same as Y, that's Y equals x minus four. That's in the slope intercept form of Y equals M X plus B. Where m is the slope and B is the Y intercept. So we can look at our Y equals x minus four. Our slope is going to be one and our y intercept or r b is going to be negative four. So let's look at our graph and we can say zero for our X value and negative for for our Y value down for and that's one point. And then our slope tells us we're going to go up 1/1 Up, And we can connect these dots to show our function our line. And now does this line pass the horizontal line test meaning we draw a horizontal line across our function and does each horizontal line only touch our function once. Yes, it does pass the horizontal line test. So it is 1 to 1 and its inverse is a function. Now let's just get to writing that inverse function. We'll start off by writing our function F of X equals x minus four. And then to get its inverse, we're going to do what I call, entering into the twilight zone. So we are going to rewrite it first. Actually, I'm going to write Y equals x minus four. And then in the twilight zone we're going to switch the X and the Y values around. So instead of y equals its X equals instead of x minus four, it's y minus four. And the goal here in the twilight zone, going through this exercise to determine the inverse is that we're going to get the Y by itself where it's happy where it truly feels fulfilled. So on the left hand side, we've got X plus four to get that Y by itself. We added affordable sides. X plus four equals Y. Oh, that y is so happy now, we can rewrite it with the Y. On the left, Y equals X plus four. And in order to get out of the twilight zone, we're going to replace that Y with the inverse function sine F to the negative one of X equals. And we bring down the right hand side X plus four. Cool, This is our inverse function. We look at our answer choices and it matches answer choice C Well done. We'll catch you on the next one

The functions in Exercises 93–95 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(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = (x - 7)/(x + 2)

Key Concepts

Inverse Functions

Verification of Inverse Functions

One-to-One Functions

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