Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.

f(x) = the quantity x minus seven divided by the quantity x plus three. and g(x) = quantity negative three x minus seven divided by quantity x minus one.

Answer 1

\[f(x)=(\frac{ x-7 }{ x+3}) and g(x)=(\frac{ -3x-7 }{ x-1})\]

Answer 2

put the g function in first then look at the f function where ever you see an x put in the whole of the g function it is untidy but simplifies to -x

Answer 3

the first x in f(x) has to be replaced by {-3x-7}/(x-1) dont forget the -7 following

Answer 4

So yes, as @jamaica25 has said for

\[\large f(g(x))\] you would take the g(x) function

\[\large g(x) = \frac{-3x - 7}{x - 1}\]

And plug that into your f(x) function wherever you see an 'x'

So for this...we have

\[\large f(x) = \frac{x - 7}{x + 3}\]

which will now become

\[\large f(g(x)) = \frac{\frac{-3x - 7}{x - 1} - 7}{\frac{-3x - 7}{x - 1} + 3}\] bad to look at now...but lets simplify it

We need a common denominator on both the top and the bottom *which are both x - 1 so

\[\large \frac{\frac{-3x - 7 - 7x + 7}{x - 1}}{\frac{-3x - 7 + 3x - 3}{x - 1}}\]

Now those \(\large x - 1\) cancel and we have

\[\large \frac{-3x - 7 - 7x + 7}{-3x - 7 + 3x - 3}\]

Combine like terms

\[\large \frac{-10x}{-10}\]

which does indeed = x

Answer 5

And it would be the same process for the g(f(x)) one...so let me know if anything trips you up :)

Answer 6

Thank you so much!!!!

Answer 7

Not a problem :)