Question

How do I find the inverse of f(x) = sqrt(x+1)?

Answer 1

To find the inverse of the function \( f(x) = \sqrt{x + 1} \), follow these steps:

1. **Replace \( f(x) \) with \( y \)**:
\[
y = \sqrt{x + 1}
\]

2. **Switch \( x \) and \( y \)**:
\[
x = \sqrt{y + 1}
\]

3. **Solve for \( y \)**:
To eliminate the square root, square both sides:
\[
x^2 = y + 1
\]

Now, isolate \( y \):
\[
y = x^2 - 1
\]

4. **Write the inverse function**:
The inverse function is:
\[
f^{-1}(x) = x^2 - 1
\]

5. **Determine the domain of the inverse**:
The original function \( f(x) = \sqrt{x + 1} \) has a domain of \( x \geq -1 \) because the expression under the square root must be non-negative. The range of \( f(x) \) is \( y \geq 0 \) since square roots are always non-negative. Therefore, the domain of \( f^{-1}(x) \) is \( x \geq 0 \).

In summary, the inverse function is:
\[
f^{-1}(x) = x^2 - 1 \quad \text{for } x \geq 0
\]