To find the inverse of the given equation [tex]\(y = 7x^2 - 10\)[/tex], let's follow the steps in detail:
1. Start with the given equation:
[tex]\[
y = 7x^2 - 10
\][/tex]
2. Swap [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Inverse of a function means swapping the roles of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
[tex]\[
x = 7y^2 - 10
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
Our goal now is to isolate [tex]\(y\)[/tex].
a. Add 10 to both sides to move the constant term to the left side:
[tex]\[
x + 10 = 7y^2
\][/tex]
b. Divide both sides by 7 to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
\frac{x + 10}{7} = y^2
\][/tex]
c. Take the square root of both sides. Remember, taking the square root gives both positive and negative solutions:
[tex]\[
y = \pm \sqrt{\frac{x + 10}{7}}
\][/tex]
Therefore, the inverse of the function [tex]\(y = 7x^2 - 10\)[/tex] is:
[tex]\[
y = \pm \sqrt{\frac{x+10}{7}}
\][/tex]
The correct answer from the given choices is:
[tex]\[
y = \pm \sqrt{\frac{x+10}{7}}
\][/tex]
