Answer :
To determine the domain and range of the function \( k(x) = -2^x \), we should first understand the behavior of the parent exponential function \( f(x) = 2^x \).
1. Domain:
- The function \( 2^x \) is defined for all real numbers \( x \). Hence, the domain of \( 2^x \) is \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \).
- The transformation \( -2^x \) does not limit the domain, so the domain of \( k(x) = -2^x \) is also \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \).
2. Range:
- For the parent function \( 2^x \), the range is all positive real numbers \( y \), i.e., \( \{ y \in \mathbb{R} \mid y > 0 \} \).
- The function \( k(x) = -2^x \) takes the output of \( 2^x \) and multiplies it by \(-1\). Therefore, it flips the graph of \( 2^x \) over the x-axis, changing all positive \( y \)-values from \( 2^x \) into negative \( y \)-values.
- As a result, the range of \( k(x) = -2^x \) is all real numbers less than or equal to 0, denoted by \( \{ y \in \mathbb{R} \mid y \leq 0 \} \).
Given these points, the correct options for the domain and range of the function \( k(x) = -2^x \) are:
- Domain: \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \)
- Range: \( \{ y \in \mathbb{R} \mid y \leq 0 \} \)
Thus, the correct option is:
- Domain: \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \)
- Range: \( \{ y \in \mathbb{R} \mid y \leq 0 \} \)
This corresponds to the first option in the given choices.
1. Domain:
- The function \( 2^x \) is defined for all real numbers \( x \). Hence, the domain of \( 2^x \) is \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \).
- The transformation \( -2^x \) does not limit the domain, so the domain of \( k(x) = -2^x \) is also \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \).
2. Range:
- For the parent function \( 2^x \), the range is all positive real numbers \( y \), i.e., \( \{ y \in \mathbb{R} \mid y > 0 \} \).
- The function \( k(x) = -2^x \) takes the output of \( 2^x \) and multiplies it by \(-1\). Therefore, it flips the graph of \( 2^x \) over the x-axis, changing all positive \( y \)-values from \( 2^x \) into negative \( y \)-values.
- As a result, the range of \( k(x) = -2^x \) is all real numbers less than or equal to 0, denoted by \( \{ y \in \mathbb{R} \mid y \leq 0 \} \).
Given these points, the correct options for the domain and range of the function \( k(x) = -2^x \) are:
- Domain: \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \)
- Range: \( \{ y \in \mathbb{R} \mid y \leq 0 \} \)
Thus, the correct option is:
- Domain: \( \{ x \in \mathbb{R} \mid -\infty < x < \infty \} \)
- Range: \( \{ y \in \mathbb{R} \mid y \leq 0 \} \)
This corresponds to the first option in the given choices.
