Find the inverse of the function 𝑓(𝑥)=4−2𝑥√x
following values:

To find the inverse of the function \( f(x) = 4 - 2x\sqrt{x} \), we will go through the following steps:

### Step 1: Express \( f(x) \) in terms of \( y \)
\[
y = 4 - 2x\sqrt{x}
\]
Now, we need to solve for \( x \) in terms of \( y \).

### Step 2: Rearrange the equation
We start by isolating \( 2x\sqrt{x} \) on one side:
\[
2x\sqrt{x} = 4 - y
\]
Divide both sides by 2:
\[
x\sqrt{x} = \frac{4 - y}{2}
\]

### Step 3: Substitute \( x = t^2 \)
Let \( x = t^2 \). Then \( \sqrt{x} = t \), and we can rewrite the equation as:
\[
t^2 t = \frac{4 - y}{2}
\]
This simplifies to:
\[
t^3 = \frac{4 - y}{2}
\]

### Step 4: Solve for \( t \)
Take the cube root of both sides:
\[
t = \sqrt[3]{\frac{4 - y}{2}}
\]

### Step 5: Recall that \( x = t^2 \)
Since \( t = \sqrt[3]{\frac{4 - y}{2}} \), substitute back to get \( x \):
\[
x = \left(\sqrt[3]{\frac{4 - y}{2}}\right)^2
\]
Thus, the inverse function is:
\[
f^{-1}(y) = \left(\sqrt[3]{\frac{4 - y}{2}}\right)^2
\]

### Final Answer:
\[
f^{-1}(y) = \left(\sqrt[3]{\frac{4 - y}{2}}\right)^2
\]

Would you like further clarification on the steps, or do you have any specific questions?

Here are five related questions:
1. How do we verify if a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. Can we find the inverse of a quadratic function? 
4. How can you graph a function and its inverse?
5. How does the domain of the original function affect its inverse?

**Tip:** When finding the inverse, try to isolate \( x \) step-by-step and ensure the function is one-to-one to have a valid inverse.

Find the inverse of the function f(x) = 4 - 2x√x

This problem guides you through finding the inverse of the function f(x) = 4 - 2x√x using algebraic steps and applying the inverse function theorem. Detailed explanations are provided to solve radical equations and determine the inverse function. Suitable for high school students in Grades 10-12.

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