Sure, let's find the inverse function [tex]\( f^{-1}(x) \)[/tex] given that [tex]\( f(x) = x^2 - 6 \)[/tex].
1. Definition of the inverse function:
The inverse function [tex]\( f^{-1} \)[/tex] is a function that, when composed with [tex]\( f \)[/tex], yields the identity function. In other words, if [tex]\( y = f(x) \)[/tex], then [tex]\( x = f^{-1}(y) \)[/tex].
2. Set up the equation:
Start with the given function:
[tex]\[
y = f(x) = x^2 - 6
\][/tex]
We need to solve this equation for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
3. Isolate [tex]\( x \)[/tex]:
Rewrite the equation in a way that allows us to isolate [tex]\( x \)[/tex]:
[tex]\[
y = x^2 - 6
\][/tex]
Add 6 to both sides:
[tex]\[
y + 6 = x^2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[
x = \pm \sqrt{y + 6}
\][/tex]
5. Determine the valid inverse function:
Since we have two possible solutions, [tex]\( x = \sqrt{y + 6} \)[/tex] and [tex]\( x = -\sqrt{y + 6} \)[/tex], we must choose the solution that aligns with the principal branch of the function typically used for defining inverse functions. In this context, the principal square root (non-negative) is generally taken:
[tex]\[
x = \sqrt{y + 6}
\][/tex]
6. State the inverse function:
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[
f^{-1}(x) = \sqrt{x + 6}
\][/tex]
However, note that the complete solution also considers the other possibility:
[tex]\[
f^{-1}(x) = -\sqrt{x + 6}
\][/tex]
Thus, the inverse [tex]\( f^{-1}(x) \)[/tex] can be written as:
[tex]\[
f^{-1}(x) = \pm \sqrt{x + 6}
\][/tex]
In typical scenarios, the principal branch considered is:
[tex]\[
f^{-1}(x) = \sqrt{x + 6}
\][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \sqrt{x + 6}
\][/tex]
