Answer :
To determine the range of the function [tex]\(f(x) = |x| + 3\)[/tex], we need to understand how the function behaves for all values of [tex]\(x\)[/tex].
1. Understand the absolute value function:
- The absolute value function [tex]\(|x|\)[/tex] outputs the non-negative value of [tex]\(x\)[/tex]. This means [tex]\(|x| \geq 0\)[/tex] for all [tex]\(x\)[/tex].
2. Examine the function [tex]\(f(x) = |x| + 3\)[/tex]:
- Since [tex]\(|x| \geq 0\)[/tex], when you add 3 to [tex]\(|x|\)[/tex], the smallest possible value of [tex]\(f(x)\)[/tex] occurs when [tex]\(|x| = 0\)[/tex].
- If [tex]\(|x| = 0\)[/tex], then [tex]\(f(x) = 0 + 3 = 3\)[/tex].
3. Determine the range:
- Since [tex]\(|x|\)[/tex] is non-negative, [tex]\(|x| \geq 0\)[/tex], any value of [tex]\(|x|\)[/tex] greater than 0 will add to 3 making [tex]\(f(x) > 3\)[/tex].
- Thus, for any real number [tex]\(x\)[/tex], [tex]\(f(x) = |x| + 3\)[/tex] will always be at least 3 or greater. Therefore, [tex]\(f(x) \geq 3\)[/tex].
Thus, the range of the function [tex]\(f(x) = |x| + 3\)[/tex] is all real numbers greater than or equal to 3.
So, the correct choice is:
[tex]\[ \{ f(x) \in R \mid f(x) \geq 3 \} \][/tex]
Hence, the answer is:
[tex]\[ \boxed{2} \][/tex]
1. Understand the absolute value function:
- The absolute value function [tex]\(|x|\)[/tex] outputs the non-negative value of [tex]\(x\)[/tex]. This means [tex]\(|x| \geq 0\)[/tex] for all [tex]\(x\)[/tex].
2. Examine the function [tex]\(f(x) = |x| + 3\)[/tex]:
- Since [tex]\(|x| \geq 0\)[/tex], when you add 3 to [tex]\(|x|\)[/tex], the smallest possible value of [tex]\(f(x)\)[/tex] occurs when [tex]\(|x| = 0\)[/tex].
- If [tex]\(|x| = 0\)[/tex], then [tex]\(f(x) = 0 + 3 = 3\)[/tex].
3. Determine the range:
- Since [tex]\(|x|\)[/tex] is non-negative, [tex]\(|x| \geq 0\)[/tex], any value of [tex]\(|x|\)[/tex] greater than 0 will add to 3 making [tex]\(f(x) > 3\)[/tex].
- Thus, for any real number [tex]\(x\)[/tex], [tex]\(f(x) = |x| + 3\)[/tex] will always be at least 3 or greater. Therefore, [tex]\(f(x) \geq 3\)[/tex].
Thus, the range of the function [tex]\(f(x) = |x| + 3\)[/tex] is all real numbers greater than or equal to 3.
So, the correct choice is:
[tex]\[ \{ f(x) \in R \mid f(x) \geq 3 \} \][/tex]
Hence, the answer is:
[tex]\[ \boxed{2} \][/tex]
