We are given the function [tex]\( k(x) = -2^x \)[/tex], and we need to determine its domain and range.
### Domain
1. Understand the exponential function: The base function [tex]\( 2^x \)[/tex] is an exponential function. Exponential functions are defined for all real numbers [tex]\( x \)[/tex].
2. Transformation impact on domain: Multiplying by [tex]\(-1\)[/tex] does not restrict the domain of the function. Therefore, the function [tex]\( k(x) = -2^x \)[/tex] is also defined for all real numbers [tex]\( x \)[/tex].
So, the domain of [tex]\( k(x) \)[/tex] is:
[tex]\[ \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \][/tex]
### Range
1. Analyze the range of the parent function: The original function [tex]\( 2^x \)[/tex] has a range of [tex]\( \{ y \in \mathbb{R} \mid y > 0 \} \)[/tex], since [tex]\( 2^x \)[/tex] is always positive for all real [tex]\( x \)[/tex].
2. Impact of the negative sign: The transformation involves multiplying the parent function by [tex]\(-1\)[/tex]. This transforms all positive output values to negative ones.
Therefore, the range of [tex]\( k(x) \)[/tex] will be:
[tex]\[ \{ y \in \mathbb{R} \mid y < 0 \} \][/tex]
However, considering that exponential functions approach zero but never actually reach zero, we need to revisit the impact thoroughly.
[tex]\[ 2^x = 0 \implies x = -\infty \][/tex]
Even as [tex]\( x \to -\infty \)[/tex], [tex]\( 2^x \to 0 \)[/tex], but will never be zero. Therefore, [tex]\( -2^x \to 0 \)[/tex] will not include zero strictly but will approach it from the negative side.
Final conclusion:
The correct range for [tex]\( k(x) \)[/tex] is:
[tex]\[ \{ y \in \mathbb{R} \mid y \leq 0 \} \][/tex]
Therefore, the correct choices are:
- Domain: [tex]\(\{ x \in \mathbb{R} \mid -\infty < x < \infty \} \)[/tex]
- Range: [tex]\(\{ y \in \mathbb{R} \mid y \leq 0 \} \)[/tex]
Thus, the correct answer is:
[tex]\[ \text{Domain: } \{ x \in \mathbb{R} \mid -\infty < x < \infty \}, \text{ Range: } \{ y \in \mathbb{R} \mid y \leq 0 \} \][/tex]
