Sure! Let's find the inverse function of [tex]\( f(x) = x^2 + 17 \)[/tex]. Here’s a step-by-step guide to solve this:
1. Start with the function:
[tex]\[
f(x) = x^2 + 17
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = x^2 + 17
\][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the equation of the inverse function:
[tex]\[
x = y^2 + 17
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
x - 17 = y^2
\][/tex]
5. To get [tex]\( y \)[/tex], take the square root of both sides:
[tex]\[
y = \sqrt{x - 17}
\][/tex]
6. Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \sqrt{x - 17}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \sqrt{x - 17}
\][/tex]
