Here are some basic inverse function problems to practice finding and understanding inverse functions. I'll include both the function notation and steps you can use to verify each inverse.

Problem Set

1. Given $f(x) = 3x + 5$, find $f^{-1}(x)$.

2. Given $f(x) = \frac{2x - 3}{5}$, find $f^{-1}(x)$.

3. Given $f(x) = x^2 - 4$ for $x \geq 0$, find $f^{-1}(x)$.

4. Given $f(x) = \frac{1}{x + 3}$, find $f^{-1}(x)$.

5. Given $f(x) = \sqrt{x + 7}$, find $f^{-1}(x)$.

Steps for Finding an Inverse

For each function, follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve for $y$ to find $f^{-1}(x)$.
4. Verify the inverse by checking $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Would you like me to solve these, or do you have questions about any specific ones?

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Additional Practice Questions

1. If $f(x) = 4x - 7$, what is $f^{-1}(x)$?
2. What is the inverse of $g(x) = \frac{x + 1}{3}$?
3. If $h(x) = x^3$, find $h^{-1}(x)$.
4. Find the inverse of $k(x) = 5 - x$.
5. Given $m(x) = |x|$, discuss why $m(x)$ might not have an inverse.

Tip:

For many functions, a quick way to check if an inverse exists is to see if the function is one-to-one. Functions that pass the horizontal line test have an inverse!