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

Using the derivative, verify that the function in the indicated exercise is always increasing or always decreasing and therefore one-to one.$$f(x)=\frac{3 x-2}{2 x+5}$$

$f^{\prime}(x)=\frac{19}{(2 x+5)^{2}} > 0$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 1

Inverse Functions

Campbell University

Baylor University

Idaho State University

Lectures

01:25

Use the derivative to help…

00:56

01:55

Using the derivative, veri…

00:40

Determine whether the foll…

02:03

Determine whether the func…

01:16

Use the first derivative t…

01:56

03:36

01:11

03:06

Suppose that $f$ is a diff…

04:13

02:52

06:48

Increasing and decreasing …

So if we want to take the derivative of this to start to show this is 1 to 1 will need to use quotient rule. So remember, caution Rule is low D Hi. Minus high. The low. Well, actually, I wrote the same thing twice. Let me do that again from so low the hi minus. Hi the low and then all over the square of what is below. So now the derivative of three X minus two is three. The derivative of two X minus five is too. So we can go ahead and strip three and two and then add everything up. So this is going to be six x plus. Yes. Butane minus six X Um, minus four. All over two X plus five squared. Now notice these here. Cancel out. And then this would become positive. So that means the numerator is going to be 19/2 X plus five squared. And so now this here, Um so two X plus five will always be zero or positive. And if we square that, then that's still zero or positive. And then if we divide anything by 19, that would also still be positive. So this is strictly larger than zero or derivative is strictly larger than zero. Uh huh, and knows how, even at the vertical ascent. Oh, that would still be going towards infinity since it's squared. So this implies f of X, always increasing, which then also implies that f of X is 1 to 1.

View More Answers From This Book

Find Another Textbook

Numerade Educator

03:01

Suppose that the rate of change at any time $t,$ of the number of bacteria i…

01:43

Find $x$.$$\log _{x} 1 / 16=-4 / 3$$

03:10

(a) Given $f(x)=3 x^{5}+2 x^{3}+2,$ show this function is one-to-one.(b)…

01:40

Solve for $x.$$$10^{\log (3 x+4)}=1$$

01:09

Determine the energy of an earthquake if its magnitude is $8.5 .$

02:31

Sketch the graph of the function defined by the given equation.$$f(x)=2.…

01:12

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

04:59

Evaluate the given integral.$$\int \ln (2 x+3) d x$$

01:19

Determine the derivative.$f(x)=e^{a x^{2}}, a$ is a constant

02:28

Determine the area of the region between the given curves.$$f(x)=x^{2}-4…