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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]
### 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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