Question

given the function f(x)=x² - 19, x ≥ 0, complete parts a through c. (a) find an equation for f⁻¹(x). (b) graph f and f⁻¹ in the same rectangular coordinate system. (c) use interval notation to give the domain and the range of f and f⁻¹. (a) find f⁻¹(x). f⁻¹(x)=□ (type an exact answer, using radicals as needed.)

Explanation:

Step1: Swap x and y

Let \( y = x^2 - 19 \) (since \( f(x)=x^2 - 19 \)). Swap \( x \) and \( y \) to get \( x = y^2 - 19 \).

Step2: Solve for y

Add 19 to both sides: \( x + 19 = y^2 \). Since the original function has \( x\geq0 \), the inverse function will have \( y\geq0 \) (because the range of \( f \) becomes the domain of \( f^{-1} \)). Take the square root of both sides: \( y=\sqrt{x + 19} \) (we take the positive square root because \( y\geq0 \)). So \( f^{-1}(x)=\sqrt{x + 19} \).

Answer:

\(\sqrt{x + 19}\)