To determine which function has a domain of all real numbers and a range that is greater than or equal to three, let's analyze each given function step by step:
1. Function [tex]\( f(x) = -x + 3 \)[/tex]
- Domain: The domain of [tex]\( f(x) \)[/tex] includes all real numbers, because there are no restrictions on the input [tex]\( x \)[/tex].
- Range: This function outputs any real number as [tex]\( x \)[/tex] varies over all real numbers, because the result can be any positive or negative number. This means the range is all real numbers, not necessarily greater than or equal to three.
2. Function [tex]\( g(x) = x^2 + 3 \)[/tex]
- Domain: The domain is all real numbers, since any real number can be squared and then added to 3.
- Range: The smallest value of [tex]\( x^2 \)[/tex] is 0, which occurs when [tex]\( x = 0 \)[/tex]. Adding 3 to this, the smallest value of [tex]\( g(x) \)[/tex] is 3. Since squaring any real number results in a non-negative value, adding 3 ensures that the function's output is always 3 or greater. Therefore, the range of [tex]\( g(x) \)[/tex] is [tex]\( y \ge 3 \)[/tex].
3. Function [tex]\( h(x) = 3^x \)[/tex]
- Domain: The domain of [tex]\( h(x) \)[/tex] includes all real numbers, as any real number can be used as an exponent for 3.
- Range: The range of [tex]\( h(x) \)[/tex] is all positive real numbers ([tex]\( y > 0 \)[/tex]), because an exponential function with base greater than 1 never reaches zero or becomes negative.
4. Function [tex]\( m(x) = |x + 3| \)[/tex]
- Domain: The domain of [tex]\( m(x) \)[/tex] includes all real numbers, since the absolute value can be computed for any real number.
- Range: The range of [tex]\( m(x) \)[/tex] is all non-negative real numbers ([tex]\( y \ge 0 \)[/tex]). The absolute value ensures the output is always [tex]\( \ge 0 \)[/tex].
From this analysis, the function that has a domain of all real numbers and a range that is greater than or equal to three is:
[tex]\[
\boxed{g(x) = x^2 + 3}
\][/tex]
