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.
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 (gh)(−6)
To find (gh)(−6), we first evaluate h(−6) and g(−6):
h(−6)=−6+5=−1g(−6)=(−6−1)(−6+2)=(−7)(−4)=28
Now, we compute (gh)(−6):
(gh)(−6)=g(−6)h(−6)=28−1
Step 2: Determine Values Not in the Domain of gh
The domain of gh 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⇒x=1x+2=0⇒x=−2
Thus, the values that are NOT in the domain of gh are x=1 and x=−2.
Final Answer
(a) 28−1
(b) Value(s) that are NOT in the domain of gh: 1, -2