Answer :
To find the inverse of the equation [tex]\( y = 2x^2 - 8 \)[/tex], we can follow these steps:
### Step 1: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
The original equation is [tex]\( y = 2x^2 - 8 \)[/tex]. To find the inverse, we begin by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2y^2 - 8 \][/tex]
### Step 2: Solve for [tex]\( y \)[/tex]
Our goal now is to solve this new equation for [tex]\( y \)[/tex]:
1. Start by isolating the term with [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ x + 8 = 2y^2 \][/tex]
2. Next, divide both sides by 2 to solve for [tex]\( y^2 \)[/tex]:
[tex]\[ \frac{x + 8}{2} = y^2 \][/tex]
3. Finally, take the square root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
Therefore, the inverse equation is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
### Step 3: Compare with given options
Now, let's compare this result with the given options:
1. [tex]\( y = \pm \sqrt{\frac{x+8}{2}} \)[/tex]
2. [tex]\( y = \frac{\pm \sqrt{x+8}}{2} \)[/tex]
3. [tex]\( y = \pm \sqrt{\frac{x}{2} + 8} \)[/tex]
4. [tex]\( y = \frac{\pm \sqrt{x}}{2} + 4 \)[/tex]
### Conclusion
The correct inverse equation matches the first option:
[tex]\[ y = \pm \sqrt{\frac{x+8}{2}} \][/tex]
Thus, the correct answer is:
[tex]\[ 1 \][/tex]
### Step 1: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
The original equation is [tex]\( y = 2x^2 - 8 \)[/tex]. To find the inverse, we begin by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2y^2 - 8 \][/tex]
### Step 2: Solve for [tex]\( y \)[/tex]
Our goal now is to solve this new equation for [tex]\( y \)[/tex]:
1. Start by isolating the term with [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ x + 8 = 2y^2 \][/tex]
2. Next, divide both sides by 2 to solve for [tex]\( y^2 \)[/tex]:
[tex]\[ \frac{x + 8}{2} = y^2 \][/tex]
3. Finally, take the square root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
Therefore, the inverse equation is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
### Step 3: Compare with given options
Now, let's compare this result with the given options:
1. [tex]\( y = \pm \sqrt{\frac{x+8}{2}} \)[/tex]
2. [tex]\( y = \frac{\pm \sqrt{x+8}}{2} \)[/tex]
3. [tex]\( y = \pm \sqrt{\frac{x}{2} + 8} \)[/tex]
4. [tex]\( y = \frac{\pm \sqrt{x}}{2} + 4 \)[/tex]
### Conclusion
The correct inverse equation matches the first option:
[tex]\[ y = \pm \sqrt{\frac{x+8}{2}} \][/tex]
Thus, the correct answer is:
[tex]\[ 1 \][/tex]
