Sure, let's analyze the properties of the functions [tex]\( f(x) = -(7)^x \)[/tex] and [tex]\( g(x) = 7^x \)[/tex] to determine the correct statement regarding their domains and ranges.
### Step 1: Determine the domain of [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -(7)^x \)[/tex] is an exponential function. Exponential functions are defined for all real numbers since you can raise a number (in this case, 7) to any power. So, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Domain of } f(x) = \text{all real numbers} \][/tex]
### Step 2: Determine the range of [tex]\( f(x) \)[/tex]
For [tex]\( f(x) = -(7)^x \)[/tex]:
- [tex]\( 7^x \)[/tex] is always positive since [tex]\( 7 \)[/tex] is a positive number and raising it to any real number exponent results in a positive number.
- The negative sign in front of [tex]\( 7^x \)[/tex] means that [tex]\( f(x) \)[/tex] will always be negative.
So, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Range of } f(x) = \text{all negative real numbers} \][/tex]
### Step 3: Determine the domain of [tex]\( g(x) \)[/tex]
The function [tex]\( g(x) = 7^x \)[/tex] is also an exponential function. Similarly, exponential functions are defined for all real numbers. So, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[ \text{Domain of } g(x) = \text{all real numbers} \][/tex]
### Step 4: Determine the range of [tex]\( g(x) \)[/tex]
For [tex]\( g(x) = 7^x \)[/tex]:
- Since [tex]\( 7^x \)[/tex] is always positive because 7 is a positive number, and raising it to any real number exponent results in a positive number.
So, the range of [tex]\( g(x) \)[/tex] is:
[tex]\[ \text{Range of } g(x) = \text{all positive real numbers} \][/tex]
### Conclusion
We determined that:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers.
- The range of [tex]\( f(x) \)[/tex] is all negative real numbers.
- The domain of [tex]\( g(x) \)[/tex] is all real numbers.
- The range of [tex]\( g(x) \)[/tex] is all positive real numbers.
This indicates that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain but different ranges. Therefore, the statement:
[tex]\[ \boxed{2. \text{ } f(x) \text{ and } g(x) \text{ have the same domain but different ranges.}} \][/tex]
is the correct one.
