Two functions f and g are defined on the set of real numbers, R, by
f:x → x\(^2\) + 2 and g:x → \(\frac{1}{x+2}\).Find the domain of (g∘f)\(^{-1}\)
Explanation
f:x → x\(^2\) + 2 and g:x → \(\frac{1}{x+2}\).
(g∘f)\(^{-1}\)
(g∘f)\(^{x^2+2}\)
g\(^{x^2+2}\) = \(\frac{1}{(x^2+2)+2}\)
(g∘f) = \(\frac{1}{(x^2+4}\)
let y = \(\frac{1}{(x^2+4}\)
y\(((x^2+4)\) = 1
\(yx^2+4y = 1\)
x\(^2\) = \(\frac{1-4y}{y}\)
x = \(\sqrt \frac{1-4y}{y}\)
(g∘f)\(^{-1}\) = \(\sqrt \frac{1-4x}{x}\)