To find (f/g)(x) for the functions f(x) = 2x² + 4x - 16 and g(x) = 4x² - 16, divide f(x) by g(x), simplify the expression by factoring and canceling out common factors, and then state the simplified result as (x² + 2x - 8) / (x + 2)(x - 2).
The student is asking how to find the quotient of the functions f(x) = 2x² + 4x - 16 and g(x) = 4x² - 16. This is done by dividing the first function by the second function, denoted as (f/g)(x). To perform this division, you would write the function f(x) over g(x), like a fraction:
(f/g)(x) = ⅛ ∴ ⅝
Step-by-Step Solution:
- Write down the functions as a fraction: (f/g)(x) = (2x² + 4x - 16) / (4x² - 16).
- Factor where possible. Both the numerator and the denominator have a common factor of 4, and the denominator is a difference of squares: (f/g)(x) = (2(x² + 2x - 8)) / (4(x + 2)(x - 2)).
- Simplify the fraction by dividing both numerator and denominator by 4: (f/g)(x) = ((x² + 2x - 8)) / ((x + 2)(x - 2)).
- Now, look for any further simplification. In this case, there's no further simplification since the numerator and the denominator do not have any more common factors.
The final result of the division (f/g)(x) remains as a rational expression:
(f/g)(x) = (x² + 2x - 8) / (x + 2)(x - 2)