To determine the range of the function [tex]\( g(x) = -2f(x) + 1 \)[/tex], let's break the problem down step-by-step.
1. Understand the Transformation:
- [tex]\( f(x) \)[/tex] is some function with an unspecified range.
- [tex]\( g(x) \)[/tex] is defined as [tex]\( -2f(x) + 1 \)[/tex].
2. Transformation Process:
- Suppose [tex]\( f(x) \)[/tex] can take any real value [tex]\( y \)[/tex], so [tex]\( f(x) = y \)[/tex].
- Now substitute [tex]\( y \)[/tex] into [tex]\( g(x) \)[/tex]: [tex]\[ g(x) = -2y + 1 \][/tex]
3. Analyze the Effect of Transformation:
- The term [tex]\( -2y \)[/tex] implies that we are scaling [tex]\( y \)[/tex] by [tex]\(-2\)[/tex].
- Adding 1 shifts the entire range of [tex]\( -2y \)[/tex] upwards by 1.
4. Identify the Range:
- If [tex]\( y \)[/tex] takes all real values (i.e., [tex]\( y \in \mathbb{R} \)[/tex]), the transformation [tex]\( -2y \)[/tex] will also take all real values (but in reverse order since it is multiplied by [tex]\(-2\)[/tex]).
- Therefore, for [tex]\( y \to \infty \)[/tex], [tex]\( -2y \to -\infty \)[/tex].
- For [tex]\( y \to -\infty \)[/tex], [tex]\( -2y \to \infty \)[/tex].
- Adding 1 to [tex]\( -2y \)[/tex] adjusts the range to still cover all real numbers but shifted by 1.
Thus, as [tex]\( y \)[/tex] spans all values from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex], the function [tex]\( g(x) = -2y + 1 \)[/tex] will cover all values from [tex]\(-\infty\)[/tex] to 1.
5. Conclusion:
- The closest answer to this range description in the given options is [tex]\( (-\infty, 1) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{C. (-\infty, 1)} \][/tex]
