Answer :
To determine the range of the function [tex]\( g(x) = -2 f(x) + 1 \)[/tex] given [tex]\( f(x) = 10^x \)[/tex], follow these steps:
1. Identify the range of [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = 10^x \)[/tex] is an exponential function. For any real number [tex]\( x \)[/tex], [tex]\( 10^x \)[/tex] will always be a positive number. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 10^x \)[/tex] approaches 0 from the positive side, and as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 10^x \)[/tex] increases without bound. Hence, the range of [tex]\( f(x) = 10^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Apply the transformation to [tex]\( f(x) \)[/tex]:
Next, consider the transformation involved in [tex]\( g(x) = -2 f(x) + 1 \)[/tex].
- First, multiply [tex]\( f(x) \)[/tex] by [tex]\(-2\)[/tex]:
- If [tex]\( f(x) \)[/tex] ranges from 0 to ∞, then [tex]\(-2 f(x) \)[/tex] will range from [tex]\( 0 \cdot -2 = 0 \)[/tex] to [tex]\( \infty \cdot -2 = -\infty \)[/tex]. Therefore, [tex]\(-2 f(x) \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 0, or [tex]\( (-\infty, 0) \)[/tex].
- Second, add 1 to all values in the range of [tex]\(-2 f(x) \)[/tex]:
- If [tex]\(-2 f(x) \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 0, adding 1 translates this range to [tex]\((- \infty + 1, 0 + 1)\)[/tex], which simplifies to [tex]\( (-\infty, 1) \)[/tex].
Thus, the range of [tex]\( g(x) = -2 f(x) + 1 \)[/tex] is [tex]\( (-\infty, 1) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(-\infty, 1)} \][/tex]
This corresponds to choice B.
1. Identify the range of [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = 10^x \)[/tex] is an exponential function. For any real number [tex]\( x \)[/tex], [tex]\( 10^x \)[/tex] will always be a positive number. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 10^x \)[/tex] approaches 0 from the positive side, and as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 10^x \)[/tex] increases without bound. Hence, the range of [tex]\( f(x) = 10^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Apply the transformation to [tex]\( f(x) \)[/tex]:
Next, consider the transformation involved in [tex]\( g(x) = -2 f(x) + 1 \)[/tex].
- First, multiply [tex]\( f(x) \)[/tex] by [tex]\(-2\)[/tex]:
- If [tex]\( f(x) \)[/tex] ranges from 0 to ∞, then [tex]\(-2 f(x) \)[/tex] will range from [tex]\( 0 \cdot -2 = 0 \)[/tex] to [tex]\( \infty \cdot -2 = -\infty \)[/tex]. Therefore, [tex]\(-2 f(x) \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 0, or [tex]\( (-\infty, 0) \)[/tex].
- Second, add 1 to all values in the range of [tex]\(-2 f(x) \)[/tex]:
- If [tex]\(-2 f(x) \)[/tex] ranges from [tex]\( -\infty \)[/tex] to 0, adding 1 translates this range to [tex]\((- \infty + 1, 0 + 1)\)[/tex], which simplifies to [tex]\( (-\infty, 1) \)[/tex].
Thus, the range of [tex]\( g(x) = -2 f(x) + 1 \)[/tex] is [tex]\( (-\infty, 1) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{(-\infty, 1)} \][/tex]
This corresponds to choice B.
