Question

Consider the function f(x)=2/5x-4

Answer 1

a.) Find the inverse of f(x) and name it g(x). Show and explain your work.
b.) Tell why or how you know that f(x) and g(x) are inverses of each other.
c.) Draw the graphs of f(x) and g(x) on the same coordinate plane. Explain what about your graph shows that the functions are inverses of each other.

Answer 2

Okay, so your function is: \(\large\color{black}{ f(x)=\frac{\LARGE 2}{\LARGE 5}x-4 }\) is that correct?

Answer 3

Yes that's correct

Answer 4

okay, so for your first step:

Replace f(x), with y, and you get. \(\large\color{black}{ y=\frac{\LARGE 2}{\LARGE 5}x-4 }\)

Answer 5

Then, switch the x and y. Like put x in place of y, and put y in place of y.

Answer 6

please write what you get... (take your time if you need to)

Answer 7

I mean put in y in place of x, (not in place of y)

Answer 8

\[x = \frac{ 2 }{ 5 } y - 4\]

Answer 9

yes

Answer 10

then solve for y.

Answer 11

add 4 to both sides (write what you get after this)

Answer 12

x = \[4 \frac{ 2 }{ 5 } y\]

Answer 13

\(\large\color{black}{ x+4=y\frac{\LARGE 2}{\LARGE 5} }\) like this.

Answer 14

Ohhh I didn't add 4 to the x

Answer 15

then multiply every thing times 5/2.

Answer 16

what happens to \(\large\color{black}{y\frac{\LARGE 2}{\LARGE 5} }\) when you multiply it times \(\large\color{black}{\frac{\LARGE 5}{\LARGE 2} }\) ?


what happens to \(\large\color{black}{x }\) when you multiply it times \(\large\color{black}{\frac{\LARGE 5}{\LARGE 2} }\) ?


what happens to \(\large\color{black}{4 }\) when you multiply it times \(\large\color{black}{\frac{\LARGE 5}{\LARGE 2} }\) ?

answer these question for me, what does each thing change to?

Answer 17

I know for #2 it would be 5/2x right?

Answer 18

yes.

Answer 19

for number 1, wouldn't 2/5 cancel with 5/2 ?

Answer 20

You'd be left with just y wouldn't you?

Answer 21

yes.

Answer 22

I got disconnected sorry for late reply.
and how about number 3?

Answer 23

So 4 would be 4/1, so you'd end up after reducing, with 10/1, right?

Answer 24

yes, it will be 10.

Answer 25

So when you multiply \(\large\color{black}{ x+4=y\frac{\LARGE 2}{\LARGE 5} }\) times 5/2, you get:

\(\large\color{black}{ \frac{\LARGE 5}{\LARGE 2} x+10=y }\)

Answer 26

so you inverse function is \(\large\color{black}{f^{-1}(x)= \frac{\LARGE 5}{\LARGE 2} x+10 }\)

Answer 27

So that inverse function would be my g(x), for part A of the problem

Answer 28

yes.

Answer 29

Now to know that your inverse function is actually the inverse function, you can:

1) Graph each function, and see if they are symmetrical about a line y=x.
2) Show that: \(\large\color{black}{f^\circ g(x)=g^\circ f(x) }\)

Answer 30

I would assume that you need to do the second thing most likely.

Answer 31

Part C is actually draw the graphs of f(x) and g(x) on the same coordinate plane. Explain what about your graph shows that the functions are inverses of each other.

Answer 32

you have:

\(\large\color{black}{f(x)= \frac{\LARGE 2}{\LARGE 5} x-4 }\)

\(\large\color{black}{g(x)= \frac{\LARGE 5}{\LARGE 2} x+10 }\)


are you good with part B, sure?

Answer 33

Yes I am. Thank you very much :). You're a life saver

Answer 34

Lol, you are lowering my status I am a Champion jk

Answer 35

Haha, that you are!

Answer 36

okay, so lets go with part C.
this is the other way (1st one that I posted)

to show that f(x) and g(x) are inverse.

basically graphing them.
I'll graph them for you.
https://www.desmos.com/calculator/orkagazgha

the \(\normalsize\color{red}{\rm red}\) is f(x)
the \(\normalsize\color{blue}{\rm blue}\) is g(x) (the inverse)
the the \(\normalsize\color{green}{\rm green}\) is y=x.

Answer 37

you see that the graphs of the \(\normalsize\color{red}{\rm f(x)}\) and \(\normalsize\color{blue}{\rm g(x)}\) a reflections (mirrors) of each other
across the \(\normalsize\color{green}{\rm y=x}\).

Answer 38

So, g(x) being a reflection of f(x), would show that, on a graph they're inverses of each other?

Answer 39

not exactly,

g(x) being a reflection of f(x) \(\normalsize\color{black}{\rm \underline{across~~the~~line~~y=x}}\), would show that, on a graph they're inverses of each other?

Answer 40

I edited your statement to be true.

Answer 41

Ohhh, okay that makes more sense now

Answer 42

If you have any questions, ask...

Answer 43

Okay :). I think I'm all good for now. Thank you so much

Answer 44

Anytime:)