Question

square root functions
instruction active
try it finding the inverse of a function
find the inverse of the function.
y = 2x² - 4
options:
y = ±√(x + 2)
y = ±√((x + 4)/2)
y = ±(√(x + 4))/2
y = ±√(x - 2)

Explanation:

Step1: Swap x and y

To find the inverse of a function \( y = 2x^2 - 4 \), we first swap \( x \) and \( y \). So we get \( x = 2y^2 - 4 \).

Step2: Solve for y

First, add 4 to both sides of the equation: \( x + 4 = 2y^2 \). Then, divide both sides by 2: \( \frac{x + 4}{2}=y^2 \). Finally, take the square root of both sides. Remember that when taking the square root, we have both positive and negative roots, so \( y=\pm\sqrt{\frac{x + 4}{2}} \), which can also be written as \( y = \pm\frac{\sqrt{x + 4}}{2} \) (since \( \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} \) for \( a\geq0,b>0 \)).

Answer:

\( y=\pm\frac{\sqrt{x + 4}}{2} \) (corresponding to the option \( y = \pm\frac{\sqrt{x + 4}}{2} \))