I have another question regarding functions, but instead if you are adding up functions that are in a radical.
f(x) and g(x) are both square root of x+5.
To find the domain of f+g, f-g, fg, and f/g, would I find the domain of each function...[-5,infinity)? And then what? How would I add each set up?
f/g would be (-5,infinity) since g(5) = 0, and you cannot divide by 0.
"adding" sets is just taking their union. AUA = A
Ok. I get it. Thanks for your help.
To find the domain of composite functions involving square roots, we need to ensure that the radicands (the expressions inside the square root) are greater than or equal to zero, as square roots are only defined for non-negative numbers.
Given that f(x) and g(x) are both the square root of (x+5), first, we need to find the domain for each function individually.
For f(x) = √(x+5), the radicand (x+5) should be greater than or equal to zero. Solving the inequality gives us:
x + 5 ≥ 0
x ≥ -5
Thus, the domain of f(x) is [-5, ∞).
Similarly, for g(x) = √(x+5), we obtain the same domain, [-5, ∞).
To find the domain of f+g, f-g, fg, and f/g, we need to consider the domain restrictions for each operation.
1. For f+g: To add two functions, the domains of both functions must be the same. Since the domains of f(x) and g(x) are both [-5, ∞), the domain of f+g would also be [-5, ∞).
2. For f-g: Similarly, when subtracting functions, the domains must be the same. Since f(x) and g(x) have the same domain, the domain of f-g would also be [-5, ∞).
3. For fg: The product of two square root functions does not have any additional domain restrictions beyond the radicands being non-negative (x+5 ≥ 0). Therefore, the domain of fg would also be [-5, ∞).
4. For f/g: When dividing functions, we need to consider that division by zero is not defined. Thus, the domain of f/g will be the overlapping area where the divisor (g(x)) is non-zero. Since g(x) = √(x+5), we need to ensure that x+5 ≠ 0. Thus, x ≠ -5. Therefore, the domain of f/g is (-5, ∞).
In summary, for f+g, f-g, fg, and f/g, all have the same domain, which is [-5, ∞), except for f/g, where the domain is (-5, ∞).