Consider functions f and g.
f(x) = x^2−16/x^2+3x−10, for x≠-5 and x≠2
g(x) = x^2−4/x^2−7x+12, for x≠3 and x ≠4
What is (f⋅g)(x)?
To find (f⋅g)(x), we need to multiply the two functions f(x) and g(x).
(f⋅g)(x) = f(x) ⋅ g(x)
First, let's write out the expressions for f(x) and g(x):
f(x) = (x^2−16)/(x^2+3x−10)
g(x) = (x^2−4)/(x^2−7x+12)
Now, let's substitute these expressions into the equation for (f⋅g)(x):
(f⋅g)(x) = [(x^2−16)/(x^2+3x−10)] ⋅ [(x^2−4)/(x^2−7x+12)]
Next, let's simplify this expression by canceling out the common factors:
(f⋅g)(x) = [(x−4)(x+4)/((x−2)(x+5))] ⋅ [(x−2)(x+2)/((x−3)(x−4))]
Since (x−4) and (x−2) appear in both the numerator and denominator of the fraction, they cancel out:
(f⋅g)(x) = [(x+4)/((x+5))] ⋅ [(x+2)/((x−3))]
Multiplying the two fractions together gives us the final expression:
(f⋅g)(x) = (x+4)(x+2)/((x+5)(x−3))
So, (f⋅g)(x) = (x+4)(x+2)/(x+5)(x−3)