Question

If r is a linear function with a slope that is not equal to zero, which of the following is always true of its inverse function?

A) Its slope is the negative reciprocal of the slope of r.
B) Its slope is the reciprocal of the slope of r.
C) Its y-intercept is the negative reciprocal of the y-intercept of r.
D) Its y-intercept is the reciprocal of the y-intercept of r.

Answer 1

let's find out...
say we have the function:
\[f(x)=mx+b\]
can you find the inverse function?

Answer 2

first step: solve y=mx+b for x

Answer 3

Ok. So that's going to be X=mY+B right?

Answer 4

ok you switch x and y around
but do you know how to solve that for Y?

Answer 5

umm i forgot how to solve it. can u pls explain me how to solve it?

Answer 6

wait so M is going to that X side right? im not sure

Answer 7

x=my+b
you want to get y by itself
so we have to undo the operations around it
notice that you have a plus b on the side with the my
subtract b on both sides

Answer 8

and yes we will later undo the operation of multiplication of m to that y by dividing both sides by m

Answer 9

ohh yeah. i remember now. so that;s going to be mx-b=y?

Answer 10

So finding the inverse function is a two step process for linear functions:

You have
x-b=my

divide both sides by m

Answer 11

x=my+b
we subtracted b on both sides
x-b=my+b-b
x-b=my+0
x-b=my
now we have one step to undo that multiplication of m onto that y
so we divide m on both sides

Answer 12

ok i got that now. then what is next?

Answer 13

well what do you have as a result of dividing m on both sides

Answer 14

x/m-b=y?

Answer 15

close
you divide both sides by m
\[\frac{x-b}{m}=\frac{my}{m} \\ \frac{x-b}{m}=y \\ y=\frac{x-b}{m} \\ y=\frac{x}{m}-\frac{b}{m} \\ y=\frac{1}{m}x-\frac{b}{m} \\ f^{-1}(x)=\frac{1}{m}x-\frac{b}{m} \\ \text{ and we have } \\ f(x)=mx+b\]

Answer 16

notice the slope of f is m and the y-intercept is b
notice the slope of inverse of f is 1/m and the y-intercept is -b/m

Answer 17

ohh yeah i got it now. understand.

Answer 18

so what is your answer then

Answer 19

is it B?

Answer 20

boom yeah
or whatever the kids are saying these days

Answer 21

haha thanks @freckles

Answer 22

np yes m and 1/m are reciprocals

Answer 23

Alright. i got it now. appreciate it @freckles