To determine the range of the function [tex]\( g(x) = f(x) + 3 \)[/tex], let's analyze the impact of adding 3 to the function [tex]\( f(x) \)[/tex].
1. Understand the range of [tex]\( f(x) \)[/tex]:
- Assume the function [tex]\( f(x) \)[/tex] can take any real number as its value, thereby having its range as [tex]\((-\infty, \infty)\)[/tex].
2. Effect of adding 3:
- When you add 3 to [tex]\( f(x) \)[/tex], each value of [tex]\( f(x) \)[/tex] is increased by 3. This operation shifts the entire range of [tex]\( f(x) \)[/tex] upwards by 3 units.
3. Determine the new range:
- Since [tex]\( f(x) \)[/tex] originally could take any real number ([tex]\((-\infty, \infty)\)[/tex]), adding 3 to each of these values does not restrict the range. Each number in [tex]\((-\infty, \infty)\)[/tex] plus 3 will still cover [tex]\( \infty \)[/tex].
Therefore, the range of [tex]\( g(x) = f(x) + 3 \)[/tex] remains [tex]\((-\infty, \infty)\)[/tex].
So, the correct answer is:
C. [tex]\((-\infty, \infty)\)[/tex]
