Question

which equation is the inverse of $y = 100 - x^2$?
$\bigcirc$ $y = \pm \sqrt{100 - x}$
$\bigcirc$ $y = 10 \pm \sqrt{x}$
$\bigcirc$ $y = 100 \pm \sqrt{x}$
$\bigcirc$ $y = \pm \sqrt{x - 100}$

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap the roles of \( x \) and \( y \) in the original equation \( y = 100 - x^2 \). So we get \( x = 100 - y^2 \).

Step2: Solve for y

Now, we solve the new equation for \( y \). First, we rearrange the equation to isolate \( y^2 \). Subtract \( x \) from both sides: \( 0 = 100 - x - y^2 \), then rearrange to \( y^2 = 100 - x \). To solve for \( y \), we take the square root of both sides. Remember that when we take the square root, we have to consider both the positive and negative roots, so \( y=\pm\sqrt{100 - x} \).

Answer:

\( y=\pm\sqrt{100 - x} \) (the first option)