If f and g are two functions defined as $f(x)=x+2, x\le0$; $g(x)=3, x\ge0,$ then the domain of $f+g$ is:

Q: If f and g are two functions defined as $f(x)=x+2, x\le0$; $g(x)=3, x\ge0,$ then the domain of $f+g$ is:

To determine the domain of the function $ f+g $, we need to consider the domains of the individual functions $ f $ and $ g $ and find their intersection, as $ f+g $ is defined only where both $ f $ and $ g $ are defined. 1. **Function $ f(x) = x + 2 $:** - The domain of $ f $ is given as $ x \le 0 $. This means $ f $ is defined for all real numbers less than or equal to 0. 2. **Function $ g(x) = 3 $:** - The domain of $ g $ is given as $ x \ge 0 $. This means $ g $ is defined for all real numbers greater than or equal to 0. To find the domain of $ f+g $, we need to find the intersection of the domains of $ f $ and $ g $. The domain of $ f $ is $ (-\infty, 0] $, and the domain of $ g $ is $ [0, \infty) $. The intersection of these two domains is the set of points where both functions are defined, which is $ \{0\} $. Now, let's analyze why Option 1 is correct and why the other options are incorrect: 1. **Option 1: $\{0\}$** - This is correct because the only point where both $ f $ and $ g $ are defined is at $ x = 0 $. Therefore, the domain of $ f+g $ is exactly $\{0\}$. 2. **Option 2: $[0, \infty)$** - This is incorrect because $ f $ is not defined for any $ x > 0 $. Therefore, $ f+g $ cannot be defined for any $ x > 0 $. 3. **Option 3: $(-\infty, \infty)$** - This is incorrect because neither $ f $ nor $ g $ is defined over the entire set of real numbers. The domain of $ f+g $ cannot be all real numbers. 4. **Option 4: $(-\infty, 0)$** - This is incorrect because $ g $ is not defined for any $ x < 0 $. Therefore, $ f+g $ cannot be defined for any $ x < 0 $. In conclusion, the only value where both functions $ f $ and $ g $ are defined is at $ x = 0 $, making Option 1 ($\{0\}$) the correct answer for the domain of $ f+g $.

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To determine the domain of the function $ f+g $, we need to consider the domains of the individual functions $ f $ and $ g $ and find their intersection, as $ f+g $ is defined only where both $ f $ and $ g $ are defined. 1. **Function $ f(x) = x + 2 $:** - The domain of $ f $ is given as $ x \le 0 $. This means $ f $ is defined for all real numbers less than or equal to 0. 2. **Function $ g(x) = 3 $:** - The domain of $ g $ is given as $ x \ge 0 $. This means $ g $ is defined for all real numbers greater than or equal to 0. To find the domain of $ f+g $, we need to find the intersection of the domains of $ f $ and $ g $. The domain of $ f $ is $ (-\infty, 0] $, and the domain of $ g $ is $ [0, \infty) $. The intersection of these two domains is the set of points where both functions are defined, which is $ \{0\} $. Now, let's analyze why Option 1 is correct and why the other options are incorrect: 1. **Option 1: $\{0\}$** - This is correct because the only point where both $ f $ and $ g $ are defined is at $ x = 0 $. Therefore, the domain of $ f+g $ is exactly $\{0\}$. 2. **Option 2: $[0, \infty)$** - This is incorrect because $ f $ is not defined for any $ x > 0 $. Therefore, $ f+g $ cannot be defined for any $ x > 0 $. 3. **Option 3: $(-\infty, \infty)$** - This is incorrect because neither $ f $ nor $ g $ is defined over the entire set of real numbers. The domain of $ f+g $ cannot be all real numbers. 4. **Option 4: $(-\infty, 0)$** - This is incorrect because $ g $ is not defined for any $ x < 0 $. Therefore, $ f+g $ cannot be defined for any $ x < 0 $. In conclusion, the only value where both functions $ f $ and $ g $ are defined is at $ x = 0 $, making Option 1 ($\{0\}$) the correct answer for the domain of $ f+g $.