To find the equation that can be simplified to determine the inverse of [tex]\( y = x^2 - 7 \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Let's go through the steps to find the inverse function:
1. Start with the given equation:
[tex]\[ y = x^2 - 7 \][/tex]
2. Add 7 to both sides of the equation to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[ y + 7 = x^2 \][/tex]
3. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{y + 7} \][/tex]
This gives us the inverse function of the original equation:
[tex]\[ x = \sqrt{y + 7} \][/tex]
Now, we need to determine which one of the given options can be simplified to match this inverse function. Let's examine each option:
1. [tex]\( x = y^2 - \frac{1}{7} \)[/tex]:
- This does not match our inverse function, as it involves subtracting [tex]\( \frac{1}{7} \)[/tex] rather than adding 7.
2. [tex]\( \frac{1}{x} = y^2 - 7 \)[/tex]:
- This implies [tex]\( x = \frac{1}{y^2 - 7} \)[/tex], which also does not match our inverse function.
3. [tex]\( x = y^2 - 7 \)[/tex]:
- If we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we have [tex]\( y = x^2 - 7 \)[/tex], which matches the original equation, indicating that this is the correct inverse relationship.
4. [tex]\( -x = y^2 - 7 \)[/tex]:
- This would imply [tex]\( x = - (y^2 - 7) \)[/tex], which is not the same as our inverse function.
Therefore, the correct option that matches the inverse function [tex]\( x = \sqrt{y + 7} \)[/tex] is:
[tex]\[ x = y^2 - 7 \][/tex]
Hence, the correct answer is the third option.
