To determine the range of the function [tex]\( g(x) \)[/tex] based on the given function [tex]\( g(x) = -2f(x) + 1 \)[/tex] where [tex]\( f(x) = 10^x \)[/tex], let's follow a detailed analysis step-by-step.
1. Understand the range of [tex]\( f(x) = 10^x \)[/tex]:
- The function [tex]\( 10^x \)[/tex] is an exponential function with a base greater than 1.
- The range of [tex]\( 10^x \)[/tex] is all positive real numbers, i.e., [tex]\( (0, \infty) \)[/tex].
2. Apply the transformation to find the range of [tex]\( g(x) = -2f(x) + 1 \)[/tex]:
- Substituting [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex], we get [tex]\( g(x) = -2 (10^x) + 1 \)[/tex].
- Let's analyze this step-by-step using the range of [tex]\( 10^x \)[/tex].
3. Calculate [tex]\(-2 \cdot f(x)\)[/tex]:
- Since [tex]\( f(x) \)[/tex] spans from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex], [tex]\( -2 f(x) \)[/tex] will span from [tex]\( -2 \cdot \infty \)[/tex] to [tex]\( -2 \cdot 0 \)[/tex].
- Therefore, [tex]\( -2 f(x) \)[/tex] spans [tex]\((-\infty, 0)\)[/tex].
4. Add 1 to the range:
- When adding 1 to the entire range of [tex]\( -2 f(x) \)[/tex], we need to shift the entire interval [tex]\((-\infty, 0)\)[/tex] upward by 1.
- This results in a new interval: [tex]\((-∞, 0) + 1\)[/tex], which simplifies to [tex]\((-∞ + 1, 0 + 1)\)[/tex].
5. Final range of [tex]\( g(x) \)[/tex]:
- Therefore, the resulting transformed range, after the shift, becomes [tex]\((-∞, 1)\)[/tex].
Conclusively, the range of the function [tex]\( g(x) = -2f(x) + 1 \)[/tex] is:
[tex]\[ (-\infty, 1) \][/tex].
