Question

1. find the inverse of the function f(x) = 16x², x ≤ 0. the inverse of f is g(x) = \boxed{}.

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = 16x^2 \) where \( x \leq 0 \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap \( x \) and \( y \), so we get \( x = 16y^2 \) with the condition that \( y \leq 0 \) (since the original function has \( x \leq 0 \), the inverse function will have \( y \leq 0 \) to maintain the domain and range relationship).

Step3: Solve for \( y \)

First, divide both sides by 16: \( \frac{x}{16}=y^2 \). Then take the square root of both sides. Since \( y \leq 0 \), we take the negative square root: \( y = -\sqrt{\frac{x}{16}}=-\frac{\sqrt{x}}{4} \) (for \( x\geq0 \), as the range of \( f(x) = 16x^2, x\leq0 \) is \( y\geq0 \), so the domain of the inverse function is \( x\geq0 \)).

Answer:

\( g(x)=-\frac{\sqrt{x}}{4}, x\geq0 \)