Which of the following is not true about the inverse of f (x) = = 3x+5
2

Answer Before defining the inverse of a function we need to have the right mental image of function.

Consider the function f(x) = 2x + 1. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. In this section it helps to think of f as transforming a 3 into a 7, and f transforms a 5 into an 11, etc.

Now that we think of f as "acting on" numbers and transforming them, we can define the inverse of f as the function that "undoes" what f did. In other words, the inverse of f needs to take 7 back to 3, and take -3 back to -2, etc.

Let g(x) = (x - 1)/2. Then g(7) = 3, g(-3) = -2, and g(11) = 5, so g seems to be undoing what f did, at least for these three values. To prove that g is the inverse of f we must show that this is true for any value of x in the domain of f. In other words, g must take f(x) back to x for all values of x in the domain of f. So, g(f(x)) = x must hold for all x in the domain of f. The way to check this condition is to see that the formula for g(f(x)) simplifies to x.

g(f(x)) = g(2x + 1) = (2x + 1 -1)/2 = 2x/2 = x.

This simplification shows that if we choose any number and let f act it, then applying g to the result recovers our original number. We also need to see that this process works in reverse, or that f also undoes what g does.

f(g(x)) = f((x - 1)/2) = 2(x - 1)/2 + 1 = x - 1 + 1 = x.

Letting f-1 denote the inverse of f, we have just shown that g = f-1.

Definition:

Let f and g be two functions. If

f(g(x)) = x and g(f(x)) = x,

then g is the inverse of f and f is the inverse of g.

Exercise 1:

Return to Contents

Finding Inverses

Example 1. First consider a simple example f(x) = 3x + 2.

The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse.

There are two steps required to evaluate f at a number x. First we multiply x by 3, then we add 2.

Thinking of the inverse function as undoing what f did, we must undo these steps in reverse order.

The steps required to evaluate f-1 are to first undo the adding of 2 by subtracting 2. Then we undo multiplication by 3 by dividing by 3.

Therefore, f-1(x) = (x - 2)/3.

Steps for finding the inverse of a function f.

Replace f(x) by y in the equation describing the function.

Interchange x and y. In other words, replace every x by a y and vice versa.

Solve for y.

Replace y by f-1(x).

Example 2.  f(x) = 6 - x/2

Step 1 y = 6 - x/2.

Step 2 x = 6 - y/2.

Step 3 x = 6 - y/2.

y/2 = 6 - x.

y = 12 - 2x.

Step 4 f-1(x) = 12 - 2x.

Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve.

Example 3.  f(x) = x3 + 2

This is the function we worked with in Exercise 1. From its graph (shown above) we see that it does have an inverse. (In fact, its inverse was given in Exercise 1.)

Step 1 y = x3 + 2.

Step 2 x = y3 + 2.

Step 3 x - 2 = y3.

(x - 2)^(1/3) = y.

Step 4 f-1(x) = (x - 2)^(1/3).

Exercise 3:

Graph f(x) = 1 - 2x3 to see that it does have an inverse. Find f-1(x). Answer

Step-by-step explanation:

pls brain list

In this figure, sin(a) = 2/3
What is the value of cos(b) ?

4x2 – 13x = 7x – 25
solve an equation with two real solutions practice

which fraction is equevleint to 12/21

Mike cuts a pie into 4 equal pieces. What is the angle measure of each piece

write 0.000066 in word form

new i wiil make you brainleist One-fourth the difference between 2/3 and 1/2.

Write an equation for the line in slope-intercept form.
Please help ASAP!!!

which comparison does the ratio 20:21 represent? ​

Paisley works on her drawings for 2 hours. She spends16 hour on each drawing. How many drawings does she work on in 2 hours?

Which set of numbers are factors of 8? *

1, 4, 5
1, 2, 5
1, 2, 4

A typical Yorkshire terrier weighs 1/6 times as many pounds as a Dalmatian. A Dalmatian weighs 54 pounds. How much does the typical Yorkshire terrier weigh?

Plz help plz help I need help

Solve for x. −3/4(x+2)=6

A rectangle length 34m x width 42m. what is the area of the rectangle

Which of the following situations describes a positive number?
A Diving underwater 4 meters.
B Going up the elevator 14 floors.
C Losing 3 points.
D) Paying $20 out of your checking account.

nancy lives on a corner lot. cutting across a lot that is 36 feet, and 8 ft longerthan the width.find the length and width if the lot

Emma has already taken 6 pages of notes on her own, and she will take 4 pages during each hour of class. Write an equation that shows the relationship between the time in class (h) and the number of pages (p).

Write one number as a fraction of another and please explain.

Is 4 5/12 a rational number

A sphere has a diameter of 11 inches. What is its surface area? Use 3.14 to approximate π. Round to the nearest hundredth if necessary.

wht do i put in the square box ?

how do you slove thia ​

what is 243.275 in expanded form​

Lamar is 5 feet 10 inches tall
. How
mary inches tallis Lamar?

Answer please.....
Thanks!!!!

Is the table a function?

what do a and b stand for in this equation. 360°÷a=b?

in the picture is my question :3 thank you so much for your help!

how many inches represent 6 feet

show if the following numbers are rational 1¾
​