Two functions f and g are defined on the set of real numbers, R, by f:x → x2 + 2 and g:x → 1x+2.Find the domain of (g∘f)−1

Two functions f and g are defined on the set of real numbers, R, by

f:x → x2 ${}^{\mathrm{<br>}}$ + 2 and g:x → 1x+2.Find the domain of (g∘f)−1

Answer Details

f:x → x2 ${}^2$ + 2 and g:x → 1x+2.

(g∘f)−1 ${}^{-1}$

(g∘f)x2+2 ${}^{{\mathrm{<br>}}^{}}$

gx2+2 ${}^{{\mathrm{<br>}}^{}}$ = 1(x2+2)+2

(g∘f) = 1(x2+4 $\frac{\mathrm{<br>}}{}$

let y = 1(x2+4 $\frac{\mathrm{<br>}}{}$

y((x2+4) $<br>$ = 1

yx2+4y=1 $\mathrm{<br>}$

x2 ${}^2$ = 1−4yy $\frac{\mathrm{<br>}}{}$

x = 1−4yy−−−−√

(g∘f)−1 ${}^{<br>}$ = 1−4xx−−−−√