We are asked to find $f(g(x))$ and check if the functions $f$ and $g$ are inverses of each other.

Given:

• $f(x) = 2x + 3$
• $g(x) = \frac{x - 3}{2}$

Step 1: Find $f(g(x))$

This means we substitute $g(x)$ into $f(x)$.

$f(g(x)) = 2\left( \frac{x - 3}{2} \right) + 3$

Simplifying: $f(g(x)) = (x - 3) + 3$ $f(g(x)) = x$

Step 2: Find $g(f(x))$

Now substitute $f(x)$ into $g(x)$.

$g(f(x)) = \frac{(2x + 3) - 3}{2}$ Simplifying: $g(f(x)) = \frac{2x}{2}$ $g(f(x)) = x$

Conclusion

Since both $f(g(x)) = x$ and $g(f(x)) = x$, the functions $f$ and $g$ are indeed inverses of each other.

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:

1. How do you find the inverse of a linear function?
2. Why is it important that both $f(g(x)) = x$ and $g(f(x)) = x$ hold true?
3. Can any function have an inverse?
4. What is the significance of the slope and intercept when finding inverses of linear functions?
5. How would you determine if two non-linear functions are inverses of each other?

Tip: A function has an inverse if it is one-to-one, meaning each output is the result of exactly one input (it passes the horizontal line test).