Question

Find the equation of the inverse of the function
f
(
x
)
=
4
x
2
+
4
with a domain restricted to
x
≥
0
as a square root function.

Answer 1

To find the inverse of the function \( f(x) = 4x^2 + 4 \) with the domain restricted to \( x \geq 0 \), we will follow these steps:

1. **Replace \( f(x) \) with \( y \):**
\[
y = 4x^2 + 4
\]

2. **Solve for \( x \) in terms of \( y \):**
\[
y - 4 = 4x^2
\]
\[
\frac{y - 4}{4} = x^2
\]
\[
x^2 = \frac{y - 4}{4}
\]
Now, take the square root of both sides. Since we are restricting \( x \) to be non-negative, we will take the positive square root:
\[
x = \sqrt{\frac{y - 4}{4}}
\]

3. **Simplify the expression:**
\[
x = \frac{\sqrt{y - 4}}{2}
\]

4. **Express the inverse function:**
Now, we switch \( x \) and \( y \) to write the inverse function:
\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2}
\]

5. **State the domain of the inverse function:**
The original function \( f(x) \) has a range starting from \( 4 \) (when \( x = 0 \)), so the domain of the inverse function is \( x \geq 4 \).

Thus, the equation of the inverse function is:
\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2} \quad \text{for } x \geq 4.
\]