Question

Confirm that f and g are inverses by showing that f(g(x))=x and g(f(x))=x f(x)=x-7/x+3 and g(x)= -3x -7/x-1

Answer 1

so, are you confused where?

Answer 2

Honestly the whole thing. I don't know why I am confused on this one but I am completely stuck

Answer 3

well,
f(x) = 2x+1
g(x) = WHATEVER

f( g(x) ) = 2(WHATEVER) +1

Answer 4

Please break it down for me for me

Answer 5

whenever is said that say, f( g(x) )

f() is really CONTAINING g(x), or one can say, f(x) is HOSTING the g(x) GUEST

Answer 6

so, every "x" or variable, in the HOST function, will be replaced by the GUEST function

Answer 7

So how can I put this into the question I have? please help me work this question out

Answer 8

do you know how to add fractions?

Answer 9

yes

Answer 10

$$\bf
f(x)=\cfrac{x-7}{x+3}\\
g(x)=\color{red}{\cfrac{-3x -7}{x-1}}\\
f(g(x)) = \cfrac{\left(\color{red}{\cfrac{-3x -7}{x-1}}\right)-7}{\left(\color{red}{\cfrac{-3x -7}{x-1}}\right)+3}
$$

Answer 11

if g(x) is the inverse of f(x)
those fractions addition will give you "x" only

Answer 12

So I wrote it down but what would you get?

Answer 13

$$f(g(x)) = \cfrac{\left(\cfrac{-3x -7}{x-1}\right)-7}{\left(\cfrac{-3x -7}{x-1}\right)+3}\\
\large \bf {\cfrac{
\frac{-3x-7-7x+7}{x-1}}{
\frac{-3x-7+3x-3}{x-1}}
}
$$

Answer 14

what would that give you?

Answer 15

-10x/x-1 -10/x-1

Answer 16

$$\bf \cfrac{
\frac{-10x}{x-1}}{
\frac{-10}{x-1}} \implies
\frac{-10x}{x-1} \times \frac{x-1}{-10}
$$

Answer 17

so, there :), is the inverse

Answer 18

and you do the same with the g( f(x) )

Answer 19

just x?

Answer 20

yes, if f( g(x) ) = x, then g(x) is the inverse, or \(\bf f^{-1}(x) \) of f(x)

Answer 21

could you please help me with the other one also?

Answer 22

recall that => \(\bf \huge \cfrac{
\frac{a}{b}}{\frac{c}{d}} \implies \frac{a}{b} \times \frac{d}{c}\)

Answer 23

so for the other one instead there be a x-1 in the denominator there would be x+3?

Answer 24

the other one is g( f(x) )
so the HOST function is g(x), and the GUEST function is f(x)
so any "x"'s in g(x) will be replaced for whatever f(x) is

Answer 25

$$\bf
f(x)=\frac{x-7}{x+3}\\
g(x)=\frac{-3x -7}{x-1}\\
g(f(x)) = \large \cfrac{-3\left(\frac{x-7}{x+3}\right) -7}{\left(\frac{x-7}{x+3}\right)-1}
$$

Answer 26

please can you tel me the answer. I am beyond confused

Answer 27

so it would be x again?

Answer 28

the exercises in themselves are meaningless, and so is the answer
what makes them useful is solving them
I mean, I can get the answer quickly, but that's not going to help you

Answer 29

the idea behind them is that you understand what happens and why and where

Answer 30

I mean, if all your teacher wanted was the answers filled in, she would have handed out to all students the answer sheet too, and that'd be all

Answer 31

I understand but please just this time I've never been this confused ever I usually get it but this one I just don't

Answer 32

well, is really just one function added to another at the "x"
so really all "x" on the HOST are just PLACEHOLDERS for the GUEST function

Answer 33

so if say,
f(x) = x + 3x + 1
g(x) = WHATEVER

f( g(x) ) = (WHATEVER)+3(WHATEVER) + 1

Answer 34

it just happens that in this one, both functions are rationals, or fractions
so you end up with a fraction over a fraction

Answer 35

so, you add the fraction atop by itself
then you add the one below by itself

then just flip the denominator as \(\bf \large \cfrac{
\frac{a}{b}}{\frac{c}{d}} \implies \frac{a}{b} \times \frac{d}{c}\)

Answer 36

which is really all I'm doing

Answer 37

$$\bf \huge {
g(f(x)) = \cfrac{-3\left(\frac{x-7}{x+3}\right) -7}{\left(\frac{x-7}{x+3}\right)-1}\\
\implies
\cfrac{
\frac{-3x+21-7x-21}{x+3}}{
\frac{x-7-x-3}{x+3}}
}
$$

Answer 38

what would you have to do after that?

Answer 39

well, notice atop, is a fraction
below, a fraction
so you flip the denominator upside-down and multiply by the numerator, like \(\bf \large \cfrac{
\frac{a}{b}}{\frac{c}{d}} \implies \frac{a}{b} \times \frac{d}{c}\)

Answer 40

-3x+21-7x-21/x+3 X x+3/x-7-x-3

Answer 41

yes, the x+3 will cancel out with each other

Answer 42

-3x-3+7x-3 so how do I connect with the x from before?

Answer 43

well, is -3x-3-7x-3

Answer 44

hmm wait

Answer 45

oh my bad. So is that the final answer?

Answer 46

the expression above is -3x+21-7x-21 => -3x-7x+21-21 => -10x
the expression below is x-7-x-3 => x-x-7-3 => -10

Answer 47

$$\bf
\large\cfrac{
\frac{-3x+21-7x-21}{x+3}}{
\frac{x-7-x-3}{x+3}} \implies
\frac{-3x+21-7x-21}{\cancel{x+3}} \times \frac{\cancel{x+3}}{x-7-x-3}\\
\frac{-10x}{-10}
$$

Answer 48

which is x

Answer 49

yes, which means g( f(x) ) = x
thus g(x) is the inverse of f(x)

Answer 50

Got the final answer thank you so much! :)