Find the inverse function of f informally. f(x)=3x+4

To find the inverse of the function \( f(x) = 3x + 4 \), follow these steps:

1. **Start with the equation for \( f(x) \):**
   \[
   y = 3x + 4
   \]
   Replace \( f(x) \) with \( y \) to make it easier to solve for \( x \).

2. **Switch \( x \) and \( y \):**
   To find the inverse, interchange the roles of \( x \) and \( y \):
   \[
   x = 3y + 4
   \]

3. **Solve for \( y \):**
   Now solve this equation for \( y \):
   \[
   x - 4 = 3y
   \]
   \[
   y = \frac{x - 4}{3}
   \]

4. **Rewrite as the inverse function:**
   The inverse function, \( f^{-1}(x) \), is:
   \[
   f^{-1}(x) = \frac{x - 4}{3}
   \]

So, the inverse of \( f(x) = 3x + 4 \) is \( f^{-1}(x) = \frac{x - 4}{3} \).

Would you like more details or have any questions?

Here are some related questions:

1. What is the domain of the inverse function?
2. How do we verify that two functions are inverses of each other?
3. Can every function have an inverse? Why or why not?
4. How does the graph of a function relate to the graph of its inverse?
5. How do transformations affect the inverse of a function?

**Tip:** A function must be one-to-one (injective) to have an inverse, meaning it passes the horizontal line test.

Find the inverse function of f informally. f(x) = 3x + 4

Learn how to find the inverse of the linear function f(x) = 3x + 4 with a step-by-step informal explanation. This problem covers key algebraic concepts related to inverse functions and solving linear equations. Suitable for students in Grades 9-11.

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