To find the inverse of the function [tex]\(f(x) = \sqrt{x+3}\)[/tex], follow these steps:
1. Replace [tex]\(f(x)\)[/tex] with [tex]\(y\)[/tex]:
[tex]\[
y = \sqrt{x+3}
\][/tex]
2. Switch [tex]\(x\)[/tex] and [tex]\(y\)[/tex] (this will help in finding the inverse function):
[tex]\[
x = \sqrt{y+3}
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
- Square both sides to eliminate the square root:
[tex]\[
x^2 = y + 3
\][/tex]
- Isolate [tex]\(y\)[/tex] by subtracting 3 from both sides:
[tex]\[
y = x^2 - 3
\][/tex]
4. Write the inverse function:
[tex]\[
f^{-1}(x) = x^2 - 3
\][/tex]
5. Determine the domain of the inverse function, considering the original function [tex]\(f(x)\)[/tex] is defined for [tex]\(x \geq -3\)[/tex]. The inverse function [tex]\(f^{-1}(x)\)[/tex] will be defined for [tex]\(x \geq -3\)[/tex]:
- The final form of the inverse function with the domain is:
[tex]\[
f^{-1}(x) = x^2 - 3 \quad \text{for} \quad x \geq -3
\][/tex]
So, the detailed solution is:
To find the inverse of the function, change [tex]\(f(x)\)[/tex] to [tex]\(y\)[/tex], switch [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and solve for [tex]\(y\)[/tex].
The resulting function can be written as [tex]\(f^{-1}(x) = x^2 - 3\)[/tex] where [tex]\(x \geq -3\)[/tex].
