Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Consider $f(x)=\frac{4 x}{x^{2}-9},$ (a) Show that this function is always decreasing.(b) Does this function have an inverse? Explain.

a. $f^{\prime}(x)=\frac{-4\left(x^{2}+9\right)}{\left(x^{2}-9\right)^{2}} < 0$ b. No, it fails the horizontal line test.

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

McMaster University

Harvey Mudd College

Lectures

01:37

A function $f$ is increasi…

01:44

Think About It A function …

02:24

(a) Prove that the functio…

Yeah, So let's go ahead and show that this is always decreasing. And then we can try to discuss whether an inverse for this function actually exist or not. So now if we take the derivative of this as one way, we can show that this will be always decreasing. So we use closure rule. So quotient Rule says it would be low, the high minus high, the low all over the square of what's below. Now, the derivative of this is for the derivative of this is two X. Now we can go ahead and multiply everything together and then try to add and subtract. So that would give us four X squared minus 36. And then that would be minus eight X squared. Uh, that would be all over X squared minus nine, squirt. And actually let me move this down a little bit. Um, so first we can go ahead and combine those that would give us negative for X squared minus 36 all over X squared minus nine squared. And then let's go ahead and factor out that negative four. So it's negative for X squared plus nine all over X squared, minus nine squared. So now, um, what we have in the denominator is always going to be greater than or equal to zero. What? We have the numerator. Well, expert plus nine is always going to be greater than equal zero. But then negative four is strictly less than zero. So we multiply all that together. That implies this is strictly less than zero. So we have that now. Now, if we were to just go ahead and use this idea to, um show that the inverse exists because it's always decreasing, there's one thing that we don't always think about, And we need to kind of consider what we're doing that, um if I were to sketch the graph of this really quickly, um, you might see why there will be no inverse for this function. So be four x over so X minus three X plus three. So if I were to sketch this over here on the side So first we have a intercept at zero, and then we have vertical ascent to set three and negative three. So 123 123 And then we also have a horizontal Jacinto at. Why is it was zero since the denominator is larger than the numerator. Let me do that. So now, uh, if you were to go through, like, figuring out, like if it's positive negative in these intervals, you actually find the graph of this function. Looks like this here. And you can see how this actually fails the horizontal line test. So this implies that, um, this has no inverse unless we restrict our domain. So mhm. You need to be careful when you're applying, um, that the derivative always being greater than zero or less than zero. Because if we have where the function is not continuous, then we might actually get something kind of weird that occurs. So that's why this one fails because we don't have continuity on the entire interval.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:22

Verify that $f$ and $g$ are inverse functions using the composition property…

01:15

The rate of increase of money at a bank is proportional to the amount invest…

00:43

Sketch the graph of the function defined by the equation $f(x)=2 x e^{-4 x}$…

01:53

Use logarithmic differentiation to find the derivative.$$y=x^{2 x}$$

01:13

Solve for $x$ in.$$8^{2 x-3}=16^{1-4 x}$$

01:04

Using the derivative, verify that the function in the indicated exercise is …

01:23

Find and classify, using the second partial derivative test, the critical po…

01:20

Determine $f^{\prime}(x)$.$$f(x)=\ln x^{3}$$

01:11

Determine $f^{\prime}(x)$.$$f(x)=\ln x^{-3}$$

02:19

Write $x^{x}$ as $e^{x \ln x}$ to find its derivative. (See Example 16 .)