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.
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Solution

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Solution Steps

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

To find (hg)(6)\left(\frac{h}{g}\right)(-6), we first evaluate h(6)h(-6) and g(6)g(-6): h(6)=6+5=1 h(-6) = -6 + 5 = -1 g(6)=(61)(6+2)=(7)(4)=28 g(-6) = (-6 - 1)(-6 + 2) = (-7)(-4) = 28 Now, we compute (hg)(6)\left(\frac{h}{g}\right)(-6): (hg)(6)=h(6)g(6)=128 \left(\frac{h}{g}\right)(-6) = \frac{h(-6)}{g(-6)} = \frac{-1}{28}

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

The domain of hg\frac{h}{g} is restricted by the values that make g(x)=0g(x) = 0. We solve for xx in the equation: g(x)=(x1)(x+2)=0 g(x) = (x - 1)(x + 2) = 0 Setting each factor to zero gives us: x1=0x=1 x - 1 = 0 \quad \Rightarrow \quad x = 1 x+2=0x=2 x + 2 = 0 \quad \Rightarrow \quad x = -2 Thus, the values that are NOT in the domain of hg\frac{h}{g} are x=1x = 1 and x=2x = -2.

Final Answer

(a) 128\boxed{\frac{-1}{28}}

(b) Value(s) that are NOT in the domain of hg\frac{h}{g}: 1, -2

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