Answer :
To address the problem, let's analyze the function [tex]\( f(x) = -7^x \)[/tex] and how it relates to the function [tex]\( g(x) = 7^x \)[/tex].
### Step-by-Step Solution:
1. Identifying the Transformation:
- The function [tex]\( f(x) = -7^x \)[/tex] can be derived from [tex]\( g(x) = 7^x \)[/tex].
- By introducing a negative sign in front of the exponential function, we are reflecting the graph of [tex]\( g(x) = 7^x \)[/tex] over the x-axis. Therefore, the answer to the first part is:
Your answer is (input a, b, c, d, e, or f): [tex]\( \textbf{e} \)[/tex]
2. Determining the Domain:
- The domain of the original function [tex]\( g(x) = 7^x \)[/tex] includes all real numbers: [tex]\( (-\infty, \infty) \)[/tex].
- The transformation to [tex]\( f(x) = -7^x \)[/tex] does not alter the values of x for which the function is defined. Thus, the domain remains all real numbers.
Your answer is (input Yes or No): [tex]\( \textbf{Yes} \)[/tex]
3. Finding the Range:
- The original function [tex]\( g(x) = 7^x \)[/tex] produces positive values for all real numbers x, so its range is [tex]\( (0, \infty) \)[/tex].
- Reflecting this function over the x-axis multiplies all outputs by -1, changing the range from positive to negative values. Hence, the range for [tex]\( f(x) = -7^x \)[/tex] becomes [tex]\( (-\infty, 0) \)[/tex].
The value of [tex]\( A \)[/tex]: [tex]\( \textbf{0} \)[/tex]
Combining the answers, we have:
1. [tex]\( e \)[/tex]
2. [tex]\( Yes \)[/tex]
3. [tex]\( 0 \)[/tex]
Thus, the detailed, step-by-step solution is completed.
### Step-by-Step Solution:
1. Identifying the Transformation:
- The function [tex]\( f(x) = -7^x \)[/tex] can be derived from [tex]\( g(x) = 7^x \)[/tex].
- By introducing a negative sign in front of the exponential function, we are reflecting the graph of [tex]\( g(x) = 7^x \)[/tex] over the x-axis. Therefore, the answer to the first part is:
Your answer is (input a, b, c, d, e, or f): [tex]\( \textbf{e} \)[/tex]
2. Determining the Domain:
- The domain of the original function [tex]\( g(x) = 7^x \)[/tex] includes all real numbers: [tex]\( (-\infty, \infty) \)[/tex].
- The transformation to [tex]\( f(x) = -7^x \)[/tex] does not alter the values of x for which the function is defined. Thus, the domain remains all real numbers.
Your answer is (input Yes or No): [tex]\( \textbf{Yes} \)[/tex]
3. Finding the Range:
- The original function [tex]\( g(x) = 7^x \)[/tex] produces positive values for all real numbers x, so its range is [tex]\( (0, \infty) \)[/tex].
- Reflecting this function over the x-axis multiplies all outputs by -1, changing the range from positive to negative values. Hence, the range for [tex]\( f(x) = -7^x \)[/tex] becomes [tex]\( (-\infty, 0) \)[/tex].
The value of [tex]\( A \)[/tex]: [tex]\( \textbf{0} \)[/tex]
Combining the answers, we have:
1. [tex]\( e \)[/tex]
2. [tex]\( Yes \)[/tex]
3. [tex]\( 0 \)[/tex]
Thus, the detailed, step-by-step solution is completed.
