To find the formula for the inverse function $f^{-1}(x)$ of the function $f(x) = 4x - 3$, we need to follow these steps:

1. Express the function $f(x)$:

$f(x) = 4x - 3$

2. Replace $f(x)$ with $y$:

$y = 4x - 3$

3. Swap $x$ and $y$ to find the inverse:

$x = 4y - 3$

4. Solve for $y$ to find $f^{-1}(x)$:

$x + 3 = 4y$ $y = \frac{x + 3}{4}$

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

5. Determine the domain of $f^{-1}(x)$:

Since $f(x) = 4x - 3$ is a linear function, it is defined for all real numbers. Therefore, $f^{-1}(x)$ is also defined for all real numbers. There are no additional restrictions on the domain of $f^{-1}(x)$ inherited from $f(x)$.

Final Answer:

• The inverse function is $f^{-1}(x) = \frac{x + 3}{4}$.
• The domain of $f^{-1}(x)$ is all real numbers $\mathbb{R}$.

Would you like more details or have any questions?

Related Questions:

1. What are the properties of inverse functions in general?
2. How do you verify that $f(x)$ and $f^{-1}(x)$ are truly inverses?
3. Can you find the inverse of a quadratic function, and how does its domain differ?
4. How does the graph of $f(x)$ relate to the graph of $f^{-1}(x)$?
5. What happens to the domain and range when finding the inverse of more complex functions?

Tip:

Always check that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ to verify your inverse function is correct.