# Question 29ecb

Oct 26, 2016

The product is $p \left(x\right) = \left(5 x + 7\right) \cdot \sqrt{2 x - 3}$, its domain is D_p=<3/2;+oo)

The quotient is $q \left(x\right) = \frac{5 x - 7}{\sqrt{2 x - 3}}$, its domain is D_q=(3/2;+oo)

#### Explanation:

First we have to calculate the domains of $f \left(x\right)$ and $g \left(x\right)$.

$f \left(x\right)$ is defined for all real numbers - ${D}_{f} = \mathbb{R}$

$g \left(x\right)$ is only defined when $2 x - 3 \ge 0$

$2 x \ge 3$

$x \ge \frac{3}{2}$

D_g=<3/2;+oo)

The construction of product and quotient is just writing correct formulas using multiplication and division:

The product is $p \left(x\right) = \left(5 x + 7\right) \cdot \left(\sqrt{2 x - 3}\right)$

The quotient is $q \left(x\right) = \frac{5 x - 7}{\sqrt{2 x - 3}}$

The product is defined for those $x$ where both factors are defined, so its domain is D_p=<3/2;+oo)

The domain of $q \left(x\right)$ is smaller, because $g \left(x\right)$ cannot be zero (it is in the denominator), so you have to exclude $\frac{3}{2}$ from the domain.

Finally D_q=(3/2;+oo)#