Question

find the inverse of each function.
$y = x^2 - 5$
$y = \sqrt{x} + 3$
$y = \frac{2x}{5}$

Explanation:

For \( y = x^2 - 5 \)

Step1: Swap \( x \) and \( y \)

\( x = y^2 - 5 \)

Step2: Solve for \( y \)

Add 5 to both sides: \( x + 5 = y^2 \)
Take square roots: \( y = \pm\sqrt{x + 5} \) (with \( x \geq - 5 \))

Step1: Swap \( x \) and \( y \)

\( x=\sqrt{y}+3 \)

Step2: Solve for \( y \)

Subtract 3: \( x - 3=\sqrt{y} \)
Square both sides: \( y=(x - 3)^2 \) (with \( x\geq3 \))

Step1: Swap \( x \) and \( y \)

\( x=\frac{2y}{5} \)

Step2: Solve for \( y \)

Multiply both sides by 5: \( 5x = 2y \)
Divide by 2: \( y=\frac{5x}{2} \)

Answer:

\( y=\pm\sqrt{x + 5} \) ( \( x\geq - 5 \) )

For \( y=\sqrt{x}+3 \)