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) = 4x - 3

Accepted Answer

Welcome back. I'm so glad you're here. We've got this function here, F of X equals x squared minus three. We want to figure out is this a 1 to 1 function and if so what is its inverse function. To start off, we are going to draw a rough sketch. So I'm writing an X axis for the horizontal and I'm writing an F of X axis for the vertical, which is the same thing as why. Now let's figure out what this would look like. We've got f of X equals x squared minus three. And let's draw some points to chart it. So we've got um X values and I'm going to write F of X values for the Y values if we stuck zero in for X. So that would look like x or zero squared minus three and zero squared is just zero. So F of X equals negative three. That's one point. So we'll go to zero on the X axis and down three values on our f of X axis. And now let's put zero in for our ffx value. So zero on the left equals x squared minus three and we'll add three double sides. So we've got three equals x squared. Take the square root of both sides to get rid of that squared for the X. And that means X equals plus or minus the square root of three. So we can put plus or minus the squirt of three for our X value when f of X equals zero. But hang on one second back in the problem it gave us a domain restriction. It said that x is going to be greater than or equal to zero. So that means our negative squared of three value can be discarded. So it's only going to be the positive square 23. Let's graph that we know that one squared is one, that two squared is four. So to get three, we would have somewhere between one and two. So we'll draw that roughly here on the X axis. And remember it was zero for the F of X value. So now I'm going to start at this um zero negative three point and draw a line going up through our square point and continuing upward to infinity. Now we can identify this functions, domain and range its domain or its X values are going to be including from zero to positive infinity and its range or it's F of X or Y values are going to be from negative three inclusive up to infinity, positive infinity for those F of X values. Now, let's see if this line passes the horizontal line test. If you recall from previous lessons. The horizontal line test tells us if the functions inverse is going to be a function and it means that that horizontal line can only pass through our function once and it does. It passes the horizontal line test. So yes, we're dealing with a 1 to 1 function that has an inverse. Now let's just figure out what that inverse is. Now to write the inverse function. We are going to take our function F of x equals x squared minus three. And we're going to rewrite it just for the purposes here of making it a little bit easier to understand why equals x squared minus three. And then we're going to go through this exercise to figure out the inverse function. And I call it entering the twilight zone here and what we do in the twilight zone is switch the X and the Y around. So instead of Y equals X equals, then it's going to be y squared instead of x squared minus three. The goal of the twilight zone is just to get that y value back by itself where it's happy. So we'll add three to both sides. So we've got X plus three equals Y squared. And now we will take the square root of both sides. Get rid of that squared for the Y. And that leaves us with plus or minus the squared of X plus three equals Y. I'm going to rewrite it with the Y value on the left just because we're used to that. So Y equals plus or minus the square root of X plus three. And now we can rewrite it to get out of the twilight zone as are inverse function F to the negative one, X replacing that Y equals. And then we keep what's on the right hand side plus or minus the square root of X plus three. Alright. That could very well be our inverse function. But let's look at a few things, we know that the inverse functions, domain and range are going to be switching around the original functions. Domain and range. So are inverse functions, domain is going to be the regular functions range. It's going to be inclusive of negative three up to positive infinity and are inverse functions range Is going to be the old functions domain which is inclusive of zero up to positive infinity. And now we could quickly graph our inverse function to see if it meets those restrictions. So we've got an X value and an F of X or Y value and we've got putting zero in for X. So that's F of X equals plus or minus square root of zero plus three or f of X equals plus or minus the square root of three. So those are our values for zero and F of X. And now let's put in zero for our f of X. So that b zero equals plus or minus the square root of X plus three. We want to get the X by itself, let's square both sides. Zero squared is still zero equals that square root is gone. Now X plus three. Subtract three from both sides. So we've got negative three equals X. So when f of x zero x is negative three. Let's plot those points. So um starting with the second one here we've got negative three for our X value. So that's three to the left on the X axis and zero. So not moving up or down on our F. Of X axis. And now we've got two more points when X equals zero. We've got positive square root of three and we've got negative squared of three. But hang on we have got a range restriction here and that says that our Y values are going to be zero up to infinity. So we can discard this second value meaning that this f of X value isn't going to have the negative square root of three. And meaning that over here in our function it's only going to be F of X equals the positive square root of X plus three. We can connect those lines from negative 30 up to the square zero squared of three going up to positive infinity. And we've graphed are inverse function. And so now we know that the inverse function is going to be the square root of X plus three and that our domain restriction is going to be from negative three up to infinity. And that matches with answer choice D. Well done. You've done it will catch 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) = 4x - 3

Key Concepts

One-to-One Functions

Inverse Functions

Verification of Inverse Functions

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