Question

Find the equation of the inverse of the function f(x)=4x2+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.

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:

### Step 1: Rewrite the function

First, we will express the function in terms of \( y \):

\[
y = 4x^2 + 4
\]

### Step 2: Solve for \( x \) in terms of \( y \)

Next, we want to isolate \( x \). We start by isolating the term involving \( x \):

\[
y - 4 = 4x^2
\]

Now, divide both sides by \( 4 \):

\[
\frac{y - 4}{4} = x^2
\]

Now, we take the square root of both sides. Since we restricted the domain to \( x \geq 0 \), we only consider the positive root:

\[
x = \sqrt{\frac{y - 4}{4}}
\]

### Step 3: Simplify the expression

We can simplify the expression further:

\[
x = \sqrt{\frac{y - 4}{4}} = \frac{\sqrt{y - 4}}{\sqrt{4}} = \frac{\sqrt{y - 4}}{2}
\]

### Step 4: Express the inverse function

Now we have expressed \( x \) in terms of \( y \). To find the inverse function, we switch \( x \) and \( y \):
\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2}
\]

### Step 5: Write the final answer

Thus, the equation of the inverse of the function \( f(x) = 4x^2 + 4 \) with the specified domain is:

\[
f^{-1}(x) = \frac{\sqrt{x - 4}}{2}
\]

### Verification (Optional)

To verify that we have found the correct inverse, we can check if \( f(f^{-1}(x)) = x \):

1. Start with \( f^{-1}(x) = \frac{\sqrt{x - 4}}{2} \).
2. Plug this into \( f \):
\[
f(f^{-1}(x)) = f\left(\frac{\sqrt{x - 4}}{2}\right) = 4\left(\frac{\sqrt{x - 4}}{2}\right)^2 + 4
\]
3. Simplifying inside the function:
\[
= 4\left(\frac{x - 4}{4}\right) + 4 = x - 4 + 4 = x
\]
Since \( f(f^{-1}(x)) = x \), our inverse is correct. Thus, the answer is:

\[
\boxed{f^{-1}(x) = \frac{\sqrt{x - 4}}{2}}
\]