To find the inverse of the equation [tex]\(y = 7x^2 - 10\)[/tex], we need to express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] and then swap the roles of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### 1. Starting with the equation:
[tex]\[y = 7x^2 - 10\][/tex]
### 2. Isolate the term containing [tex]\(x\)[/tex]:
[tex]\[y + 10 = 7x^2\][/tex]
### 3. Solve for [tex]\(x^2\)[/tex]:
[tex]\[x^2 = \frac{y + 10}{7}\][/tex]
### 4. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[x = \pm \sqrt{\frac{y + 10}{7}}\][/tex]
At this point, [tex]\(x\)[/tex] is expressed in terms of [tex]\(y\)[/tex]. To find the inverse function, we swap [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
### 5. Swap [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[y = \pm \sqrt{\frac{x + 10}{7}}\][/tex]
Thus, the inverse of the equation [tex]\(y = 7x^2 - 10\)[/tex] is:
[tex]\[y = \pm \sqrt{\frac{x + 10}{7}}\][/tex]
Now we compare this result with the given options:
1. [tex]\(y = \frac{\pm \sqrt{x+10}}{7}\)[/tex]
2. [tex]\(y = \pm \sqrt{\frac{x+10}{7}}\)[/tex]
3. [tex]\(y = \pm \sqrt{\frac{x}{7} + 10}\)[/tex]
4. [tex]\(y = \frac{\pm \sqrt{x}}{7} \pm \frac{\sqrt{10}}{7}\)[/tex]
The correct answer is:
[tex]\[y = \pm \sqrt{\frac{x + 10}{7}}\][/tex]
So, the correct option is:
[tex]\[y = \pm \sqrt{\frac{x+10}{7}}\][/tex]
