find the given fuctions f and g, find f*g and state the domain of f*g, f(x)=�ã2x-5,g(x)=|x-1|
To find f * g, we need to calculate the product of the given functions f(x) = 2x - 5 and g(x) = |x - 1|.
Substituting f(x) and g(x) into the expression f * g, we have:
f * g = (2x - 5) * |x - 1|
To simplify the expression, we consider two cases for the absolute value function g(x) = |x - 1|:
Case 1: x - 1 ≥ 0
In this case, |x - 1| equals (x - 1). Therefore, we have:
f * g = (2x - 5) * (x - 1)
Case 2: x - 1 < 0
In this case, |x - 1| simplifies to -(x - 1), which equals (1 - x). Hence, we have:
f * g = (2x - 5) * (1 - x)
Now, let's multiply the terms to get the final expression for f * g:
Case 1:
f * g = 2x² - 2x - 5x + 5
Case 2:
f * g = 2x - 2x² - 5 + 5x
Simplifying further, we get:
Case 1:
f * g = 2x² - 7x + 5
Case 2:
f * g = -2x² + 3x + 5
The domain of f * g is the set of values of x for which the expression f * g is defined. Since f * g is a polynomial function, there are no restrictions on the domain. Therefore, the domain of f * g is all real numbers (or (-∞, ∞)).