To find the inverse function \( f^{-1}(x) \) of the given function \( f(x) = 1 + 4x \), follow these steps:

### Step 1: Express the function as \( y = f(x) \)
\[
y = 1 + 4x
\]

### Step 2: Swap \( y \) and \( x \)
To find the inverse, swap \( x \) and \( y \):
\[
x = 1 + 4y
\]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \) to find the inverse function:
\[
x - 1 = 4y
\]
\[
y = \frac{x - 1}{4}
\]

### Step 4: Express the inverse function
The inverse function is:
\[
f^{-1}(x) = \frac{x - 1}{4}
\]

So, the inverse function \( f^{-1}(x) \) is:
\[
\boxed{f^{-1}(x) = \frac{x - 1}{4}}
\]

Would you like further details or have any questions about this solution?

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Here are some related questions you might find interesting:
1. How would you verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \)?
2. What is the domain and range of \( f(x) = 1 + 4x \)?
3. How would the inverse function change if \( f(x) = ax + b \) for different values of \( a \) and \( b \)?
4. How can you find the inverse of a quadratic function?
5. If \( f(x) = 1 + 4x \), what is the composition \( f(f^{-1}(x)) \)?
6. How does the graph of \( f(x) \) compare to the graph of \( f^{-1}(x) \)?
7. What happens to the inverse function if \( f(x) \) is not a one-to-one function?
8. Can every function have an inverse? Why or why not?

**Tip:** When finding the inverse of a function, ensure that the function is one-to-one. This means each output is associated with exactly one input, making the inverse function well-defined.

Learn how to find the inverse function of f(x) = 1 + 4x step-by-step. This problem covers concepts of inverse functions and linear functions in mathematics. Suitable for high school students.

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