Questions: Refer to the functions r, p, and q. Find the function (r/q)(x) and write the domain in interval notation.
r(x) = 4x
p(x) = x^2 + 2x
q(x) = sqrt(8-x)
Part: 0 / 2
Part 1 of 2
(r/q)(x) =
Transcript text: Refer to the functions $r, p$, and $q$. Find the function $\left(\frac{r}{q}\right)(x)$ and write the domain in interval notation.
\[
r(x)=4 x \quad p(x)=x^{2}+2 x \quad q(x)=\sqrt{8-x}
\]
Part: $0 / 2$
Part 1 of 2
\[
\left(\frac{r}{q}\right)(x)=
\]
$\square$
Solution
Solution Steps
Step 1: Define the Function (qr)(x)
To find the function (qr)(x), we need to divide the function r(x) by the function q(x).
Given:
r(x)=4xq(x)=8−x
The function (qr)(x) is defined as:
(qr)(x)=q(x)r(x)=8−x4x
Step 2: Determine the Domain of (qr)(x)
The domain of (qr)(x) is determined by the restrictions on q(x), since division by zero is undefined and the square root function requires non-negative arguments.
Square Root Restriction: The expression under the square root, 8−x, must be non-negative:
8−x≥0⟹x≤8
Denominator Restriction: The denominator 8−x cannot be zero:
8−x=0⟹8−x=0⟹x=8
Combining these conditions, the domain of (qr)(x) is:
x≤8andx=8