To find the inverse of the given function [tex]\( f(x) = x^2 - 6 \)[/tex], we will follow a series of mathematical steps.
1. First, we replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]. This gives us:
[tex]\[
y = x^2 - 6
\][/tex]
2. Next, we need to find the inverse. To do this, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = y^2 - 6
\][/tex]
3. Now, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. Add 6 to both sides of the equation:
[tex]\[
x + 6 = y^2
\][/tex]
4. Take the square root of both sides to solve for [tex]\( y \)[/tex]. Remember that taking the square root of both sides will give us both the positive and negative roots:
[tex]\[
y = \pm \sqrt{x + 6}
\][/tex]
Therefore, the inverse of the function [tex]\( f(x) = x^2 - 6 \)[/tex] is [tex]\( y = \sqrt{x + 6} \)[/tex] and [tex]\( y = -\sqrt{x + 6} \)[/tex]. In a more compact notation, we can write the inverse as:
[tex]\[
f^{-1}(x) = \pm \sqrt{x + 6}
\][/tex]
So, the set of inverse functions is:
[tex]\[
f^{-1}(x) = -\sqrt{x + 6} \quad \text{and} \quad f^{-1}(x) = \sqrt{x + 6}
\][/tex]
