Math Snap

STEP 1

1. We are given two functions $g(x) = \frac{1}{2}x + 9$ and $h(x) = (x+1)(x-1)$.
2. We need to perform operations on these functions as specified in parts a through e.
3. Simplification of expressions is required where possible.

STEP 2

1. Calculate $h(x) - g(x)$.
2. Calculate $h \times g(x)$.
3. Calculate $\frac{h}{g}(x)$.
4. Calculate the composition $g \circ h(x)$.
5. Calculate the composition $h \circ g(x)$.

STEP 3

Calculate $h(x) - g(x)$.
First, find $h(x)$:
$$h(x) = (x+1)(x-1) = x^2 - 1$$ Now subtract $g(x)$:
$$h(x) - g(x) = (x^2 - 1) - \left(\frac{1}{2}x + 9\right)$$ Simplify the expression:
$$h(x) - g(x) = x^2 - \frac{1}{2}x - 10$$

STEP 4

Calculate $h \times g(x)$.
Multiply $h(x)$ and $g(x)$:
$$h(x) \times g(x) = (x^2 - 1) \times \left(\frac{1}{2}x + 9\right)$$ Distribute:
$$= x^2 \times \frac{1}{2}x + x^2 \times 9 - 1 \times \frac{1}{2}x - 1 \times 9$$ $$= \frac{1}{2}x^3 + 9x^2 - \frac{1}{2}x - 9$$

STEP 5

Calculate $\frac{h}{g}(x)$.
Divide $h(x)$ by $g(x)$:
$$\frac{h(x)}{g(x)} = \frac{x^2 - 1}{\frac{1}{2}x + 9}$$ This expression is already simplified as much as possible.

STEP 6

Calculate the composition $g \circ h(x)$.
Substitute $h(x)$ into $g(x)$:
$$g(h(x)) = g(x^2 - 1)$$ $$= \frac{1}{2}(x^2 - 1) + 9$$ Simplify:
$$= \frac{1}{2}x^2 - \frac{1}{2} + 9$$ $$= \frac{1}{2}x^2 + \frac{17}{2}$$

Calculate the composition $h \circ g(x)$.
Substitute $g(x)$ into $h(x)$:
$$h(g(x)) = h\left(\frac{1}{2}x + 9\right)$$ $$= \left(\frac{1}{2}x + 9 + 1\right)\left(\frac{1}{2}x + 9 - 1\right)$$ Simplify:
$$= \left(\frac{1}{2}x + 10\right)\left(\frac{1}{2}x + 8\right)$$ Expand:
$$= \left(\frac{1}{2}x\right)^2 + \left(\frac{1}{2}x\right)(8) + 10\left(\frac{1}{2}x\right) + 80$$ $$= \frac{1}{4}x^2 + 4x + 5x + 80$$ $$= \frac{1}{4}x^2 + 9x + 80$$