Question

Part 1. Using the two functions listed below, insert numbers in place of the letters a, b, c, and d so that f(x) and g(x) are inverses.

f(x)=(x+a)/b

g(x)=cx−d

Part 2. Show your work to prove that the inverse of f(x) is g(x).

Part 3. Show your work to evaluate g(f(x)).

Part 4. Graph your two functions on a coordinate plane. Include a table of values for each function. Include 5 values for each function. Graph the line y = x on the same graph.

Answer 1

do you think you can help?

Answer 2

@hero ? Please :)

Answer 3

\(f(x) = \dfrac{x + a}{b}\)

\(g(x) = cx - d\)

Answer 4

yup thats it :)

Answer 5

And in order for \(f(x)\) and \(g(x)\) to be inverses:
\(f(g(x)) = x\)
and
\(g(f(x)) = x\)

Answer 6

ok....

Answer 7

What if

a = d
b = c

Answer 8

that cant be possible because a can only equal a for example 1 cant be 2, 1 is 1

Answer 9

Basically, from the work I just did, the only way for f(x) and g(x) to be inverses, that is the condition that must be met

a = d
b = c

Answer 10

So for example, if \(f(x) = \dfrac{x + 3}{4}\) then \(g(x) = 4x - 3\)

Answer 11

aaaaahhh.... now i get it

Answer 12

Because

\(f(g(x)) = \dfrac{(4x - 3) + 3}{4} = \dfrac{4x}{4} =x \)

Answer 13

so is that the answer for part 1?

Answer 14

oh, ok so what about part 2?

Answer 15

\(g(f(x)) = 4\left(\dfrac{x + 3}{4}\right) - 3 = x\)

Answer 16

is that also the work?

Answer 17

Actually, I didn't notice all the different parts that you have.

Answer 18

ohhh, sorry :(

Answer 19

Okay, so to find the inverse of f(x) for example:

\(f(x) = \dfrac{x + 3}{4}\)

Answer 20

Let y = f(x):

\(y = \dfrac{x + 3}{4}\)

Answer 21

Then swap x and y:
\(x = \dfrac{y + 3}{4}\)

Answer 22

And isolate y again:

\(4x = y + 3\)

\(4x - 3 = y\)

Then replace y with \(g(x)\)
\(4x - 3 = g(x)\)

Answer 23

For part 4, you can use desmos.com for plotting graphs.

Answer 24

yeah , i use it all the time :)

Answer 25

Could you rewrite part 3?

Answer 26

\(f(g(x)) = \dfrac{(4x - 3) + 3}{4} = \dfrac{4x}{4} =x\)

\(g(f(x)) = 4\left(\dfrac{x + 3}{4}\right) - 3 = x\)

Answer 27

And then 2?

Answer 28

I just went over part 2 above. It starts with:
"to find the inverse of f(x)"

Answer 29

so wait is part 1 the same as 3?

Answer 30

Part 1 is to find values for a, b, c, and d.

Answer 31

AHH... OK!

Answer 32

what is a, b, c, and d?

Answer 33

This is the point where you should be able to think for yourself and go over this information on your own to figure out the values of a, b, c, d. Start from the very beginning and read over this discussion and try to figure out the values on your own.

Answer 34

is a and d 3
and b and d 4?

Answer 35

where's c?

Answer 36

the d in the first one was supposed to be a c

Answer 37

Re-type the whole thing you are proposing here.

Answer 38

is a and c 4?
and is b and d 3?

Answer 39

ok i got it:
a - 3
b - 4
c - 4
d - 3

Answer 40

@Hero