You’ve provided two images. The first image defines four functions, and the second image presents a composition of functions question. Let’s break it down:

Functions provided:

1. $f(x) = \frac{3}{2}x + 1$
2. $g(x) = x^2 - 5x$
3. $h(x) = \frac{2x - 1}{33}$
4. $j(x) = 3x + 1$

Question:

Find $(f \circ g)(x)$, which means the composition of $f(x)$ and $g(x)$, or in simpler terms, $f(g(x))$.

Solution:

To find $(f \circ g)(x) = f(g(x))$, we substitute $g(x)$ into $f(x)$.

1. Recall: $f(x) = \frac{3}{2}x + 1$ So, to compute $f(g(x))$, replace $x$ in $f(x)$ with $g(x)$, which is $x^2 - 5x$.

2. Substitution: $f(g(x)) = \frac{3}{2}(x^2 - 5x) + 1$

3. Now, distribute the $\frac{3}{2}$ to each term inside the parentheses: $f(g(x)) = \frac{3}{2}x^2 - \frac{15}{2}x + 1$

Thus, the final expression for $(f \circ g)(x)$ is: $(f \circ g)(x) = \frac{3}{2}x^2 - \frac{15}{2}x + 1$

Would you like a further explanation or to go through additional examples?

5 Related Questions:

1. What is $(g \circ f)(x)$?
2. How do you find $(h \circ j)(x)$?
3. What happens if you compose $f(x)$ with itself, $(f \circ f)(x)$?
4. How would you solve $f(g(2))$?
5. How do you find the inverse of the function $f(x)$?

Tip:

When composing functions, always substitute one function into the other carefully, and simplify step by step to avoid errors.