Question

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

Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown.

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Answer 1

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

### Step 1: Replace \( f(x) \) with \( y \)
We start by rewriting the equation:
\[
y = 4x^2 + 4
\]

### Step 2: Solve for \( x \) in terms of \( y \)
Next, we need to solve this equation for \( x \). First, we will isolate the \( x^2 \) term:
\[
y - 4 = 4x^2
\]
\[
x^2 = \frac{y - 4}{4}
\]

Now we take the square root of both sides to solve for \( x \). Since we are restricting \( x \) to be non-negative (because the original function is defined for \( x \geq 0 \)), we only take the positive square root:
\[
x = \sqrt{\frac{y - 4}{4}}
\]

### Step 3: Simplify the expression
We can simplify the right side:
\[
x = \frac{\sqrt{y - 4}}{2}
\]

### Step 4: Write the inverse function
Now, we express the inverse function \( f^{-1}(y) \) by replacing \( y \) with \( x \). Thus we have:
\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2}
\]

### Final Result
The equation of the inverse function is:
\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2}
\]

This completes the derivation of the inverse function as a square root function.