To find the inverse of the function [tex]\( y = x^2 - 10x \)[/tex], we need to follow several steps. Let's go through these steps one by one:
1. Rewrite the function: The given function is [tex]\( y = x^2 - 10x \)[/tex].
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]: To do this, we'll first rewrite the function in a way that isolates [tex]\( x \)[/tex]. Notice that we have a quadratic equation. We can rewrite it as:
[tex]\[
y = x^2 - 10x
\][/tex]
We need to complete the square:
[tex]\[
y = x^2 - 10x
\][/tex]
[tex]\[
y = (x^2 - 10x + 25) - 25
\][/tex]
[tex]\[
y = (x - 5)^2 - 25
\][/tex]
Now we have:
[tex]\[
y + 25 = (x - 5)^2
\][/tex]
3. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
\sqrt{y + 25} = |x - 5|
\][/tex]
This gives us two possible solutions because the absolute value function results in both positive and negative solutions:
[tex]\[
x - 5 = \sqrt{y + 25} \quad \text{or} \quad x - 5 = -\sqrt{y + 25}
\][/tex]
Solving for [tex]\( x \)[/tex] in both cases:
[tex]\[
x = 5 + \sqrt{y + 25} \quad \text{or} \quad x = 5 - \sqrt{y + 25}
\][/tex]
4. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]: The inverse function [tex]\( y \)[/tex] should be expressed in terms of [tex]\( x \)[/tex]. We can now rewrite these as the inverse functions:
[tex]\[
y = \sqrt{x - 25} - 5 \quad \text{or} \quad y = -\sqrt{x - 25} - 5
\][/tex]
5. Match with given options:
- Option 1: [tex]\( y = \pm \sqrt{x - 25} - 5 \)[/tex]
- Option 2: [tex]\( y = \pm \sqrt{x - 25} + 5 \)[/tex]
- Option 3: [tex]\( y = \pm \sqrt{x + 25} - 5 \)[/tex]
- Option 4: [tex]\( y = \pm \sqrt{x + 25} + 5 \)[/tex]
Notice that our derived inverse functions match the forms found in Option 1.
Thus, the correct inverse of [tex]\( y = x^2 - 10x \)[/tex] is given by:
[tex]\[
y = \pm \sqrt{x - 25} - 5
\][/tex]
