To find the inverse of the equation [tex]\( y = x^2 - 7 \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
y = x^2 - 7
\][/tex]
2. Isolate [tex]\( x^2 \)[/tex] by adding 7 to both sides:
[tex]\[
y + 7 = x^2
\][/tex]
3. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
\sqrt{y + 7} = x \quad \text{or} \quad -\sqrt{y + 7} = x
\][/tex]
Since we are interested in expressing [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], we can combine both solutions to find the inverse function. However, here we need to match one of the given options closely.
4. Check the choices provided:
- [tex]\( x = y^2 - \frac{1}{7} \)[/tex]
- [tex]\( \frac{1}{x} = y^2 - 7 \)[/tex]
- [tex]\( x = y^2 - 7 \)[/tex]
- [tex]\( -x = y^2 - 7 \)[/tex]
The correct form of [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] we derived is related to [tex]\( \sqrt{y + 7} \)[/tex] and [tex]\(-\sqrt{y + 7} \)[/tex], yet none of these reflect the direct conversion. Considering these offered equations as options for some transformation, aligning the terms:
3. Check the best transformation:
Direct substitution to reach something logical by examining transformation patterns:
- Checking [tex]\( x = y^2 - 7 \)[/tex], examined:
If [tex]\( x \approx \sqrt{y+7} \approx y^2 - 7\)[/tex]
However offered rule, it solves clean linear term placement approximated:
Transformed correctly [tex]\(x=y^{2}-7\)[/tex] solving structure
Thus,
The equation correctly simplified for suitable resulting the inverse function as:
Option is:
[tex]\[ x = y^2 - 7. \][/tex]
Thus, the inverse function of the original equation [tex]\( y = x^2 - 7 \)[/tex] best transformable and correspondense simplified best match result is:
[tex]\[ x = y^2 - 7 \][/tex]
is correct.
