To find the inverse function of [tex]\( f(x) = \sqrt{4x + 6} \)[/tex], follow these steps:
1. Rewrite the equation using [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt{4x + 6}
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{4y + 6}
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
a. Square both sides to eliminate the square root:
[tex]\[
x^2 = 4y + 6
\][/tex]
b. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x^2 - 6}{4}
\][/tex]
4. State the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = 2 + \sqrt{10}
\][/tex]
Therefore, the inverse function is:
[tex]\[
f^{-1}(x) = 2 + \sqrt{10}
\][/tex]
