Question

I'm try to solve this last part of my question and I just can't prove why the functions are inverses of each other on a graph.

Answer 1

be prepared to do a little algebra

Answer 2

also, your inverse is wrong

Answer 3

some how you change \(x-4\) in to \(4-x\)

Answer 4

x - 4 is not the same as 4 - x

Answer 5

so lets start at the beginning

Answer 6

ok

Answer 7

or just agree it should be \[f^{-1}(x)=\frac{x-4}{-7}\] which would be more natural to write as either \[f^{-1}(x)=-\frac{x-4}{7}\] or \[f^{-1}(x)=\frac{4-x}{7}\]

Answer 8

you did all the steps correct, but when you subtracted \(4\) from both sides you wrote \(4-x\) instead of \(x-4\)

Answer 9

btw you should do this in your head for a line

Answer 10

\[f(x)=-7x+4\] means "multiply by \(-7\) then add \(4\)
inverse will say "subtract 4, then divide by -7"

Answer 11

oh you have to do something else for sure!!

Answer 12

plugging in 1 for the first one accomplishes nothing other than telling you that \(f(1)=-7+4=-3\)
it does mean that \(f^{-1}(-3)=7\) if we did it right

Answer 13

what you have to do is to show that \[f(f^{-1}(x))=x\]

Answer 14

that is where the algebra comes in

Answer 15

it is not as hard as it looks

Answer 16

\[f(f^{-1}(x))=f(\frac{4-x}{7})\]is the first step

Answer 17

then since \[f(\spadesuit)=-7\spadesuit+4\] you cut and paste and get \[f(\frac{4-x}{7})=-7\times \frac{4-x}{7}+4\]

Answer 18

since you know you will end up with just \(x\) you should expect a raft of cancellation

Answer 19

the 7 goes

Answer 20

to give \[-(4-x)+4\] then a tiny bit more algebra gives \(x\)

Answer 21

if you want to be ambitious you can also show that \[f^{-1}(f(x))=x\] as well

Answer 22

ok now i just read your question and it says "on a graph"
you can graph both \(y=-7x+4\) and \(y=\frac{1}{7}x+\frac{4}{7}\) and see that they are symmetric with respect to \(y=x\)

Answer 23

ok i made a mistake there
one is \[y=-7x+4\] the other is \[y=\frac{1}{7}x-\frac{4}{7}\]

Answer 24

plug in multiples of 7
it is still going to suck though
here is a nice picture
http://www.wolframalpha.com/input/?i=inverse+-7x%2B4