To find the inverse of the function \( f(x) = 7\sqrt{x} + 2 \), follow these steps:

### 1. Replace \( f(x) \) with \( y \):
\[
y = 7\sqrt{x} + 2
\]

### 2. Swap \( x \) and \( y \) to start solving for the inverse:
\[
x = 7\sqrt{y} + 2
\]

### 3. Isolate \( \sqrt{y} \):
\[
x - 2 = 7\sqrt{y}
\]

### 4. Divide by 7:
\[
\frac{x - 2}{7} = \sqrt{y}
\]

### 5. Square both sides to remove the square root:
\[
\left(\frac{x - 2}{7}\right)^2 = y
\]

### 6. Therefore, the inverse function is:
\[
f^{-1}(x) = \left(\frac{x - 2}{7}\right)^2
\]

Would you like more details on any specific step or have any questions? Here are some related questions to expand on this topic:

1. What are the domain and range of the inverse function?
2. How can you verify that \( f^{-1}(x) \) is correct by composition of functions?
3. How does squaring affect the domain and range of \( f^{-1}(x) \)?
4. Can you graph both \( f(x) \) and \( f^{-1}(x) \)?
5. What happens to the function if we change the constant in \( f(x) \)?

**Tip:** When finding the inverse, always ensure that the function is one-to-one (bijective), so that the inverse exists.

Learn how to find the inverse of the function f(x) = 7√x + 2 using algebraic methods. This solution breaks down key concepts like radical functions and inverse operations, making it suitable for high school students in Grades 10-12.

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