Question
Select the functions \( f(x)=\sqrt{x+5} \) and \( g(x)=\sqrt{5-x} \). Select the Operation \( f+g \) and Combine Functions box, and slowly slide the \( x \)-slider to the right to graph the function \( (f+g)(x) \). Check the Show Domain box. What is the domain of the function? Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure. The domain of \( (f+g)(x) \) is . (Type your answer in interval notation.)
Ask by Sandoval Johnston. in the United States
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The domain of (f+g)(x) is [-5, 5].
Solution
The function (f+g)(x) = √(x+5) + √(5-x) is defined when both square roots are defined. That requires:
1. x + 5 ≥ 0, which implies x ≥ -5.
2. 5 - x ≥ 0, which implies x ≤ 5.
Thus, the domain of (f+g)(x) is the intersection of x ≥ -5 and x ≤ 5, which is [-5, 5].
So, the domain of (f+g)(x) is [-5, 5].
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Simplify this solution Extra Insights
To find the domain of the function \((f+g)(x) = f(x) + g(x) = \sqrt{x+5} + \sqrt{5-x}\), we first need to determine the restrictions imposed by each square root function. For \(f(x) = \sqrt{x+5}\), the expression \(x + 5 \geq 0\) leads us to \(x \geq -5\). For \(g(x) = \sqrt{5-x}\), the expression \(5 - x \geq 0\) gives us \(x \leq 5\). Combining these two inequalities, we find that: - The lower bound is \(x \geq -5\). - The upper bound is \(x \leq 5\). Thus, the domain of \((f+g)(x)\) can be expressed in interval notation as \([-5, 5]\).
