Finding h(x) given f(x) and g(x)
The problem is asking to find h(x) = g(x) + f(x) - g(x) · f(x), given that f(x) = x / (x + 3) and g(x) = -3 / (2x + 3).
First, let's find g(x) + f(x):
g(x) + f(x) = (-3 / (2x + 3)) + (x / (x + 3)).
We need to find a common denominator to combine these fractions. The least common denominator (LCD) will be the product of both denominators since they have no factors in common: (2x + 3)(x + 3).
Now let's put each fraction over this common denominator:
g(x) + f(x) = (-3)(x + 3) / ((2x + 3)(x + 3)) + (x)(2x + 3) / ((2x + 3)(x + 3)).
Combine the terms:
g(x) + f(x) = [(-3x - 9) + (2x² + 3x)] / ((2x + 3)(x + 3)).
Simplify:
g(x) + f(x) = (2x² - 3x - 9) / ((2x + 3)(x + 3)).
Next, let's find g(x) · f(x):
g(x) · f(x) = [(-3 / (2x + 3))] · [(x / (x + 3))].
Multiply the numerators and the denominators:
g(x) · f(x) = (-3x) / ((2x + 3)(x + 3)).
Now, let's find h(x) by subtracting g(x) · f(x) from g(x) + f(x):
h(x) = g(x) + f(x) - g(x) · f(x)
= (2x² - 3x - 9) / ((2x + 3)(x + 3)) - (-3x) / ((2x + 3)(x + 3)).
Since the denominators are the same, we can combine the numerators:
h(x) = (2x² - 3x - 9 + 3x) / ((2x + 3)(x + 3)).
Simplify the numerator:
h(x) = (2x² - 9) / ((2x + 3)(x + 3)).
This is the simplified form of h(x). As for restrictions on the domain, since we cannot divide by zero, both denominators (2x + 3) and (x + 3) must be non-zero. Hence, x cannot be -3/2 or -3. Thus, the domain of h(x) is all real numbers except x ≠ -3/2 and x ≠ -3.
