To find the inverse of the function [tex]\( y = 5x^2 + 10 \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Here is a step-by-step solution:
1. Start with the equation [tex]\( y = 5x^2 + 10 \)[/tex].
2. To isolate the [tex]\( x \)[/tex] term, first subtract 10 from both sides of the equation:
[tex]\[
y - 10 = 5x^2
\][/tex]
3. Next, divide both sides by 5 to further isolate [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{y - 10}{5} = x^2
\][/tex]
4. To solve for [tex]\( x \)[/tex], take the square root of both sides. Remember that taking the square root gives two possible solutions (positive and negative):
[tex]\[
x = \pm \sqrt{\frac{y - 10}{5}}
\][/tex]
However, in the context of finding an inverse function, we typically choose one branch based on additional information about the original function. Assuming that [tex]\( x \geq 0 \)[/tex] is a reasonable consideration to keep the function monotonically increasing, we simplify it to:
[tex]\[
x = \sqrt{\frac{y - 10}{5}}
\][/tex]
Therefore, our inverse function would be:
[tex]\[
y^{-1}(x) = \sqrt{\frac{x - 10}{5}}
\][/tex]
Now we look at the given options to see which one matches our derived inverse function:
- [tex]\( x = 5y^2 + 10 \)[/tex]
- [tex]\( \frac{1}{y} = 5x^2 + 10 \)[/tex]
- [tex]\( -y = 5x^2 + 10 \)[/tex]
- [tex]\( y = \frac{1}{5}x^2 + \frac{1}{10} \)[/tex]
None of the given options directly represent the inverse we derived. Therefore, it seems like the problem may contain a typo or a misunderstanding.
In summary, the correct inverse function derived is:
[tex]\[ y^{-1}(x) = \sqrt{\frac{x - 10}{5}} \][/tex]
This is the function that correctly represents the inverse of [tex]\( y = 5x^2 + 10 \)[/tex].
