Question: if $h(x)=2x+3$ and $g(x)=5x-4$ write a simplified expression for $hg(x)$
1. The problem asks to find the simplified expression for the composition $hg(x)$, which means $h(g(x))$.
2. Recall the definition of function composition: $hg(x) = h(g(x))$ means we substitute $g(x)$ into $h(x)$.
3. Given $h(x) = 2x + 3$ and $g(x) = 5x - 4$, substitute $g(x)$ into $h$:
$$hg(x) = h(g(x)) = 2(g(x)) + 3$$
4. Substitute $g(x) = 5x - 4$:
$$hg(x) = 2(5x - 4) + 3$$
5. Distribute the 2:
$$hg(x) = 2 \times 5x - 2 \times 4 + 3 = 10x - 8 + 3$$
6. Simplify the constants:
$$hg(x) = 10x - 5$$
7. Therefore, the simplified expression for $hg(x)$ is:
$$hg(x) = 10x - 5$$