To determine the range of the function [tex]\( g(x) = f(x) + 3 \)[/tex]:
1. Understand the range of [tex]\( f(x) \)[/tex]:
- Assume that the function [tex]\( f(x) \)[/tex] is given and has a range that spans all real numbers, i.e., [tex]\( (-\infty, \infty) \)[/tex].
2. Transformation Applied to [tex]\( f(x) \)[/tex]:
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( f(x) \)[/tex] shifted up by 3 units. Algebraically, this is expressed as [tex]\( g(x) = f(x) + 3 \)[/tex].
3. Effect of Adding 3:
- When you add 3 to each value of [tex]\( f(x) \)[/tex], you shift the entire range of [tex]\( f(x) \)[/tex] three units upwards.
- This means if [tex]\( y \)[/tex] is any value in the range of [tex]\( f(x) \)[/tex], then [tex]\( y + 3 \)[/tex] will be the corresponding value in the range of [tex]\( g(x) \)[/tex].
4. Range of [tex]\( g(x) \)[/tex]:
- Since [tex]\( f(x) \)[/tex] spans all real numbers ([tex]\( y \)[/tex] can be any real number), [tex]\( f(x) + 3 \)[/tex] will span all real numbers shifted up by 3 units.
- Mathematically, if [tex]\( f(x) \in (-\infty, \infty) \)[/tex], then [tex]\( f(x) + 3 \in (3, \infty) \)[/tex].
Therefore, the range of [tex]\( g(x) = f(x) + 3 \)[/tex] is [tex]\((3, \infty)\)[/tex], and the correct answer is:
D. [tex]\((3, \infty)\)[/tex]
