Question

f(g(x)) type question

Answer 1

What did I do wrong?

Answer 2

you know that to verify that \(\large\color{slate}{ f^{-1}(x) }\) is indeed the inverse of \(\large\color{slate}{ f(x) }\), you would need to show that:

\(\large\color{black}{ \color{red}{f(}\color{blue}{f^{-1}(x)}\color{red}{)} =\color{blue}{f^{-1}(\color{red}{f(x)})} }\)

Answer 3

Yes

Answer 4

The question is asking for me to show that f(g(x)) and g(f(x)) both equal x I think. Can I just show that they equal each other?

Answer 5

I think it would be the same exact thing.

Answer 6

I mean if each of them is equal to x, then they are equal to each other, but I think you actually need to show that each is equal to x.

Answer 7

The idea is to show that they reflect each other over the graph of x, saying over a line of y=x.

Answer 8

swagmaster47 is not asking if both are equal
is asking if they're inverse of each other

Answer 9

and you'd do it as SolomonZelman said
by getting of f(g(x)), you should end up with "x"
and the other way around

Answer 10

on the 3rd line, your multiplication on the numerator above, went awry
\(\bf f(g(x))=\cfrac{\frac{-3x-7}{x-1}-7}
{\frac{-3x-7}{x-1}+3}\implies
\cfrac{\frac{-3x-7+[(x-1)\cdot -7]}{\cancel{ x-1}}}{\frac{-3x-7+[3(x-1)]}{\cancel{ x-1}}}
\\ \quad \\
\implies \cfrac{-3x-7-7x+7}{-3x-7+3x-3}\implies ?\)

Answer 11

-10x/10 then x

Answer 12

I mean -10x/-10

Answer 13

and then when I get to
g(f(x)) = (-3( (x-7)/(x + 3) )-7)/( ( (x-7)/(x + 3) )-1) I multiply by (x+3)/(x+3) ?

Answer 14

would that look like (-3( (x-7) )-7(x+3) )/( (x-7) -(x+3) )

Answer 15

well... those (x-3) cancel out because is a division of fractions
and \(\bf \cfrac{\frac{a}{b}}{\frac{c}{{\color{blue}{ d}}}}\implies \cfrac{a}{b}\cdot \cfrac{{\color{blue}{ d}}}{c}\) would cancel each otehr out in the multiplication

Answer 16

hmm x+3 I meant.. anyhow, so they cancel each other out

Answer 17

Oh, so it looks like this? (-3( (x-7) - 7)/( x - 7 ) - 1)

Answer 18

I would just remove b and d in your diagram right?

Answer 19

If b = d

Answer 20

yes, because in this case b = d yes

Answer 21

Ok I think I get it! thanks.

Answer 22

yw

Answer 23

Wait what happens to the -3 coeffecient?

Answer 24

Nvm, got the right answer