To determine the range of the function [tex]\( g(x) = -2 \cdot 10^x + 1 \)[/tex], we need to understand how transformations affect the range of the original function [tex]\( f(x) = 10^x \)[/tex].
1. Find the range of [tex]\( f(x) = 10^x \)[/tex]:
- As [tex]\( x \)[/tex] takes any real number, [tex]\( 10^x \)[/tex] will always be positive.
- Specifically, [tex]\( 10^x \)[/tex] spans from [tex]\( 0 \)[/tex] to [tex]\( \infty \)[/tex] (but never actually reaches [tex]\( 0 \)[/tex]).
- So, the range of [tex]\( f(x) = 10^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Applying the transformation [tex]\( g(x) = -2 \cdot f(x) + 1 \)[/tex]:
- Let's first consider the transformation [tex]\( -2 \cdot f(x) \)[/tex].
- Since [tex]\( f(x) \)[/tex] is positive for all [tex]\( x \)[/tex], multiplying it by [tex]\( -2 \)[/tex] will flip all values to be negative.
- Therefore, [tex]\( -2 \cdot 10^x \)[/tex] will range from [tex]\( -\infty \)[/tex] to [tex]\( 0 \)[/tex] (but never actually reach [tex]\( 0 \)[/tex]).
- Next, add 1 to the result:
- Adding 1 to the interval [tex]\( (-\infty, 0) \)[/tex] shifts the entire interval upwards by 1 unit.
- This transforms the range to [tex]\( (-\infty, 1) \)[/tex].
Based on the above transformations, the range of [tex]\( g(x) = -2 \cdot 10^x + 1 \)[/tex] is [tex]\( (-\infty, 1) \)[/tex].
Therefore, the correct answer is:
A. [tex]\((- \infty, 1)\)[/tex]
