To determine the domain and range of the function [tex]\( k(x) = -2^x \)[/tex], let's analyze it step-by-step.
1. Domain:
- The domain of an exponential function is the set of all possible input values ([tex]\(x\)[/tex]) that the function can take. For [tex]\( k(x) = -2^x \)[/tex], we can input any real number into the function and it will produce a valid output.
- Therefore, the domain of [tex]\( k(x) = -2^x \)[/tex] is all real numbers, which can be expressed as [tex]\( \{x \in R \mid -\infty < x < \infty\} \)[/tex].
2. Range:
- To find the range, we need to determine the possible values that [tex]\( k(x) \)[/tex] can output. The function [tex]\( f(x) = 2^x \)[/tex] is the standard exponential function, which always yields positive values for all real numbers [tex]\( x \)[/tex].
- Since [tex]\( k(x) \)[/tex] introduces a negative sign, [tex]\( k(x) = -2^x \)[/tex] will always yield negative values or zero because it negates the positive output of [tex]\( 2^x \)[/tex].
- Therefore, the range of [tex]\( k(x) = -2^x \)[/tex] includes all real numbers less than or equal to zero. This can be expressed as [tex]\( \{y \in R \mid y \leq 0\} \)[/tex].
Combining these observations, the correct domain and range of the function [tex]\( k \)[/tex] are:
- Domain: [tex]\( \{x \in R \mid -\infty < x < \infty\} \)[/tex]
- Range: [tex]\( \{y \in R \mid y \leq 0\} \)[/tex]
So, the correct answer is:
Domain: [tex]\( \{x \in R \mid -\infty < x < \infty\} \)[/tex]
Range: [tex]\( \{y \in R \mid y \leq 0\} \)[/tex]
