Find the domain of (f X g)(x) if
f(x) = 2 / x-3 and g(x) = 1 / x+4.
I know the domain for g(x) is x cannot be -4 but I do not know what else to do.
To find the domain of (f X g)(x), we need to consider the domains of both f(x) and g(x), and find any restrictions that apply to both functions.
For f(x), the expression f(x) = 2 / (x-3) is undefined when the denominator, (x-3), equals zero. So, to find the domain of f(x), we need to determine the value of x that makes the denominator zero:
x - 3 = 0
x = 3
Therefore, the function f(x) is defined for all values of x except x = 3. So, the domain of f(x) is (-∞, 3) U (3, +∞).
Now, for g(x), the expression g(x) = 1 / (x+4) is undefined when the denominator, (x+4), equals zero. So we need to solve for x:
x + 4 = 0
x = -4
Therefore, the function g(x) is defined for all values of x except x = -4. So, the domain of g(x) is (-∞, -4) U (-4, +∞).
To find the domain of (f X g)(x), we need to find the values of x that make both f(x) and g(x) defined. As we identified earlier, the domain for g(x) is x ≠ -4, and the domain for f(x) is x ≠ 3.
When combining both domains, we need to find the intersection of the two sets. The only value that satisfies both conditions is x ≠ 3 and x ≠ -4.
Therefore, the domain of (f X g)(x) is (-∞, -4) U (-4, 3) U (3, +∞).