To find the inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = \sqrt{4x + 6} \)[/tex], follow these steps:
1. Start with the given function:
[tex]\[
f(x) = \sqrt{4x + 6}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt{4x + 6}
\][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This step reflects the fact that for the inverse function, the roles of the dependent and independent variables are reversed:
[tex]\[
x = \sqrt{4y + 6}
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
x^2 = 4y + 6
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[
x^2 - 6 = 4y
\][/tex]
6. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x^2 - 6}{4}
\][/tex]
7. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex], which denotes the inverse function:
[tex]\[
f^{-1}(x) = \frac{x^2 - 6}{4}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{x^2 - 6}{4}
\][/tex]
