Let's find the inverse of the function [tex]\( y = 2x^2 - 8 \)[/tex] step by step.
1. Start with the given equation:
[tex]\[
y = 2x^2 - 8
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
First, add 8 to both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[
y + 8 = 2x^2
\][/tex]
Next, divide both sides by 2:
[tex]\[
\frac{y + 8}{2} = x^2
\][/tex]
Now, take the square root of both sides. Remember to account for both the positive and negative roots:
[tex]\[
x = \pm \sqrt{\frac{y + 8}{2}}
\][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
4. Check the given options:
Let's compare our result with the options given:
- Option 1: [tex]\( y = \pm \sqrt{\frac{x + 8}{2}} \)[/tex]
- Option 2: [tex]\( y = \frac{\pm \sqrt{x + 8}}{2} \)[/tex]
- Option 3: [tex]\( y = \pm \sqrt{\frac{x}{2} + 8} \)[/tex]
- Option 4: [tex]\( y = \frac{\pm \sqrt{x}}{2} + 4 \)[/tex]
We see that our derived inverse function matches exactly with Option 1:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
So, the correct inverse equation is:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
