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Find the inverse of the function [tex]\( y = 2x^2 + 2 \)[/tex].

A. [tex]\( y = \pm \sqrt{x - 2} \)[/tex]

B. [tex]\( y = \pm \sqrt{\frac{1}{2} x - 1} \)[/tex]

C. [tex]\( y = \pm \sqrt{2x^2 - 4} \)[/tex]

D. [tex]\( y = \pm \sqrt{\frac{1}{2} x - 1} \)[/tex]


Sagot :

To find the inverse of the function [tex]\( y = 2x^2 + 2 \)[/tex], we need to follow these steps.

### Step-by-Step Solution:

1. Write the function:
[tex]\[ y = 2x^2 + 2 \][/tex]

2. To find the inverse, swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2y^2 + 2 \][/tex]

3. Solve for [tex]\( y \)[/tex]:
First, subtract 2 from both sides:
[tex]\[ x - 2 = 2y^2 \][/tex]

Then divide both sides by 2:
[tex]\[ \frac{x - 2}{2} = y^2 \][/tex]

4. Take the square root of both sides:
[tex]\[ y = \pm \sqrt{\frac{x - 2}{2}} \][/tex]

5. Simplify the expression:
The two solutions are:
[tex]\[ y = \sqrt{\frac{x - 2}{2}} \quad \text{and} \quad y = -\sqrt{\frac{x - 2}{2}} \][/tex]

Hence, the inverse functions are:

[tex]\[ y = \sqrt{\frac{x - 2}{2}} \quad \text{and} \quad y = -\sqrt{\frac{x - 2}{2}} \][/tex]

These solutions correspond to:
[tex]\[ \pm \sqrt{\frac{x - 2}{2}} \][/tex]

Thus, the correct choice from the given options is:
[tex]\[ y = \pm \sqrt{\frac{x - 2}{2}} \][/tex]

However, it appears that there is a slight error in the provided options. Correcting the inverse function form, we get:

[tex]\[ y = \pm \sqrt{\frac{x - 2}{2}} \][/tex]

So the actual correct form which wasn't perfectly matched in the given options is:

[tex]\[ \pm \sqrt{\frac{x - 2}{2}} \][/tex]
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