Question

Consider the function f(x) = √5x + 15 for the domain [-3, ∞).Find f-¹(x), where f-1 is the inverse of f.Also state the domain of f¹ in interval notation.

Answer 1

Answer: We have to find the inverse function of the following:

$f(x)=\sqrt{5x+15}\rightarrow[-3,\infty)\Rightarrow(1)$

The Inverse of the (1) is as follows:

$\begin{gathered} f(x)=y=\sqrt{5x+15} \\ \\ \text{ Switch: }x\text{ and }y\text{ and solve for }y(x)\text{ } \\ \\ \\ y=\sqrt{5x+15}\rightarrow x=\sqrt{5y+15} \\ \\ \\ x^2=5y+15 \\ \\ \\ x^2-15=5y \\ \\ \\ y=\frac{x^2}{5}-\frac{15}{5} \\ \\ \\ \\ y=f^{-1}(x)=\frac{x^2}{5}-3 \\ \\ \\ f^{-1}(x)=\frac{x^{2}}{5}-3 \end{gathered}$

The domain of the inverse function is:

$x\in(-\infty,+\infty)$