Questions: Suppose that the functions h and g are defined as follows. h(x) = x + 5 g(x) = (x - 1)(x + 2) (a) Find (h/g)(-6). (b) Find all values that are NOT in the domain of h/g. If there is more than one value, separate them with commas.

Suppose that the functions h and g are defined as follows.

h(x) = x + 5
g(x) = (x - 1)(x + 2)

(a) Find (h/g)(-6).
(b) Find all values that are NOT in the domain of h/g.

If there is more than one value, separate them with commas.

Transcript text: Suppose that the functions $h$ and $g$ are defined as follows. \[ \begin{array}{l} h(x)=x+5 \\ g(x)=(x-1)(x+2) \end{array} \] (a) Find $\left(\frac{h}{g}\right)(-6)$. (b) Find all values that are NOT in the domain of $\frac{h}{g}$. If there is more than one value, separate them with commas.

Solution

Solution Steps

Step 1: Evaluate $\left(\frac{h}{g}\right)(-6)$

To find $\left(\frac{h}{g}\right)(-6)$, we first evaluate $h(-6)$ and $g(-6)$: \[ h(-6) = -6 + 5 = -1 \] \[ g(-6) = (-6 - 1)(-6 + 2) = (-7)(-4) = 28 \] Now, we compute $\left(\frac{h}{g}\right)(-6)$: \[ \left(\frac{h}{g}\right)(-6) = \frac{h(-6)}{g(-6)} = \frac{-1}{28} \]

Step 2: Determine Values Not in the Domain of $\frac{h}{g}$

The domain of $\frac{h}{g}$ is restricted by the values that make $g(x) = 0$. We solve for $x$ in the equation: \[ g(x) = (x - 1)(x + 2) = 0 \] Setting each factor to zero gives us: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \] Thus, the values that are NOT in the domain of $\frac{h}{g}$ are $x = 1$ and $x = -2$.