Question

If `f(x)=2x-7` and `fog(x)=4x+3` find `(gof)^(-1)(x)`?

Answer 1

`(gcircf)^(-1)(x)=(x+9)/4`

Explanation:

Given
`color(white)("XXX")f(x)=2x-7`
and since
`color(white)("XXX")fcircg(x)=f(g(x))=2 * g(x) -7`

But we are also told that
`color(white)("XXX")fcircg(x)=4x+3`

So
`color(white)("XXX")2 * g(x)-7 = 4x+3`

`color(white)("XXX")rarr 2 * g(x)=4x+10`

`color(white)("XXX")rarr g(x)=2x+5`

Therefore
`color(white)("XXX")gcircf(x)=(g(f(x))`
`color(white)("XXXXXXxX")=2 * f(x) +5`
`color(white)("XXXXXXxX")=2(2x-7)+5`
`color(white)("XXXXXXxX")=4x-14+5`
`color(white)("XXXXXXxX")=4x-9`

For simplicity, let `y=gcircf(x)`
and we are asked to find `(gcircf)^(-1)(x)=y^(-1)`

Given an equation with `y` defined in terms of `x`
the easiest way to find the inverse, `y^(-1)` is to solve the equations
to define `x` in terms of `y`
then replace `x` with `y^(-1)` and `y` with `x`:

We have
`color(white)("XXX")y=4x-9`

`color(white)("XXX")4x=y+9`

`color(white)("XXX")x=(y+9)/4`

then doing the replacement
`color(white)("XXX")y^(-1)=(x+9)/4`

that is
`color(white)("XXX")(gcircf)^(-1)(x)=(x+9)/4`