Question

Consider the following functions
f(x)= (7x+8)/(x+3) and g(x)= (3x-8)/(7-x)
(a) Find g(g(x))
(b) Find g(f(x))
(c) Determine whether the functions f and g are inverses of each other.

Answer 1

bunch o algebra for this one

Answer 2

is the first one \(g(g(x))\) or is it \(f(g(x))\) ?

Answer 3

f(g(x)) sorry

Answer 4

@satellite73

Answer 5

ok ready?

Answer 6

yes

Answer 7

\[f(g(x))=f\left(\frac{3x-8}{7-x}\right)\] is the first step

Answer 8

then where you see an \(x\) in \(f(x)=\frac{7x+8}{x+3}\) replace it by \(\frac{3x-8}{7-x}\) to get
\[f(g(x))=\frac{7\left(\frac{3x-8}{7-x}\right)+8}{\frac{3x-8}{7-x}+3}\]

Answer 9

then simplify the compound fraction by multiplying top and bottom by \(7-x\)

Answer 10

the first step is to write
\[\frac{7(3x-8)+8(7-x)}{3x-8+3(7-x)}\]then multiply out, cancel etc

Answer 11

Oh ok gotcha!

Answer 12

if you do it carefully, you will see that there is a whole mess of cancellation
you will just get \(x\) which shows that
\[f(g(x))=x\] which is a good indication that \(f\) and \(g\) are inverse functions