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Which equation can be simplified to find the inverse of [tex]\( y = 5x^2 + 10 \)[/tex] ?

A. [tex]\( x = 5y^2 + 10 \)[/tex]
B. [tex]\( \frac{1}{y} = 5x^2 + 10 \)[/tex]
C. [tex]\( -y = 5x^2 + 10 \)[/tex]
D. [tex]\( y = \frac{1}{5}x^2 + \frac{1}{10} \)[/tex]

Sagot :

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].