Question

If f(x) = 3x + 10 and g(x) = x² –1, then (fog)^–1 is equal to ___________.

Answer 1

Given: f(x) = 3x + 10 and g(x) = x² –1

$$\begin{gathered} fog\left(x\right)=f\left(g\left(x\right)\right) \\ =f\left(x^2-1\right) \\ =3\left(x^2-1\right)+10 \\ =3x^2-3+10 \\ =3x^2+7 \\ \mathrm{Thus},fog\left(x\right)=3x^2+7 \\ \Rightarrow y=3x^2+7 \\ \Rightarrow 3x^2=y-7 \\ \Rightarrow x^2=\frac{y-7}{3} \\ \Rightarrow x=\pm \sqrt{\frac{y-7}{3}} \\ fo{g}^{-1}\left(x\right)=\pm \sqrt{\frac{x-7}{3}} \end{gathered}$$

Hence, (fog)^–1 is equal to $\underline{\pm \sqrt{\frac{x-7}{3}}}$.