Recall that we must excluxe the values that make any denominator equal to zero.
See solution.
Practice makes perfect
Let's consider the following pair of rational functions.
f(x) = p(x)/q(x) and
g(x) = r(x)/s(x)
For the domain of f(x) we exclude the values that make q(x)=0. Similarly, for the domain of g(x) we exclude the values that make s(x)=0. Next, let's compute the product and quotient of these two functions.
Product of Rational Functions
Let's find f(x)g(x).
f(x)* g(x) &= p(x)/q(x)* r(x)/s(x)
&= p(x)r(x)/q(x)s(x)
For the domain of f(x)* g(x) we must exclude the values that make q(x)s(x)=0. By the Zero Product Property, we get the following two equations.
q(x) = 0
s(x) = 0
In other words, for the domain of the product of two rational functions, we must exclude the values that make q(x)=0 and the values that make s(x)=0.
Quotient of Rational Functions
Let's find f(x)÷ g(x).
f(x)/g(x) = p(x)/q(x)/r(x)/s(x) = p(x)s(x)/q(x)r(x)
For the domain of f(x)÷ g(x) we must exclude the values that make q(x)r(x)=0 and s(x)=0. Again, the Zero Product Property gives us the following two equations.
q(x) = 0
r(x) = 0
In conclusion, for the domain of the quotient of two rational functions, we must exclude the values that make q(x)=0, the values that make r(x)=0, and the values that make s(x)=0.
