To determine the inverse of the given function [tex]\( y = 7x^2 - 10 \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Let's work through the algebraic steps to find this inverse:
1. Start with the given equation:
[tex]\[ y = 7x^2 - 10 \][/tex]
2. Isolate the term containing [tex]\( x \)[/tex] on one side. Begin by adding 10 to both sides of the equation:
[tex]\[ y + 10 = 7x^2 \][/tex]
3. Next, divide both sides of the equation by 7 to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{y + 10}{7} = x^2 \][/tex]
4. To solve for [tex]\( x \)[/tex], take the square root of both sides. Remember to include both the positive and negative square roots:
[tex]\[ x = \pm \sqrt{\frac{y + 10}{7}} \][/tex]
Thus, the equation describing the inverse function is:
[tex]\[ y = \pm \sqrt{\frac{x + 10}{7}} \][/tex]
Among the provided choices, this matches with:
[tex]\[ y = \pm \sqrt{\frac{x + 10}{7}} \][/tex]
So, the correct equation for the inverse of [tex]\( y = 7x^2 - 10 \)[/tex] is:
[tex]\[ y = \pm \sqrt{\frac{x + 10}{7}} \][/tex]
