2022 WAEC Further Mathematics Theory Two functions f and g are defined on the set of real numbers, R, by...

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)\(^{-1}\)

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)\(^{-1}\) = \(\sqrt \frac{1-4x}{x}\)

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