Which statement best describes the domain and range of [tex]$f(x)=-(7)^x$[/tex] and [tex]$g(x)=7^x$[/tex]?
A. [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] have the same domain and the same range.
B. [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] have the same domain but different ranges.
C. [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] have different domains but the same range.
D. [tex]$f(x)$[/tex] and [tex]$g(x)$[/tex] have different domains and different ranges.
Answer (2)
Determine the domain of f ( x ) = − ( 7 ) x and g ( x ) = 7 x , which are both all real numbers.
Determine the range of f ( x ) = − ( 7 ) x , which is all negative real numbers.
Determine the range of g ( x ) = 7 x , which is all positive real numbers.
Conclude that f ( x ) and g ( x ) have the same domain but different ranges, so the answer is f ( x ) and g ( x ) have the same domain but different ranges.
Explanation
Analyzing the Functions We are asked to compare the domains and ranges of the functions f ( x ) = − ( 7 ) x and g ( x ) = 7 x . Let's analyze each function separately.
Domain of f(x) First, let's consider the domain of f ( x ) = − ( 7 ) x . The function 7 x is defined for all real numbers x , since we can raise 7 to any power. Multiplying by -1 does not change the domain. Therefore, the domain of f ( x ) is all real numbers.
Domain of g(x) Next, let's consider the domain of g ( x ) = 7 x . As mentioned before, 7 x is defined for all real numbers x . Therefore, the domain of g ( x ) is all real numbers.
Range of f(x) Now, let's find the range of f ( x ) = − ( 7 ) x . Since 7 x is always positive for any real number x , − ( 7 ) x will always be negative. Therefore, the range of f ( x ) is all negative real numbers, or ( − ∞ , 0 ) .
Range of g(x) Next, let's find the range of g ( x ) = 7 x . Since 7 x is always positive for any real number x , the range of g ( x ) is all positive real numbers, or ( 0 , ∞ ) .
Comparing Domains and Ranges Comparing the domains, we see that both f ( x ) and g ( x ) have the same domain: all real numbers. Comparing the ranges, we see that f ( x ) has a range of all negative real numbers, while g ( x ) has a range of all positive real numbers. Therefore, the ranges are different.
Final Answer Therefore, the statement that best describes the domain and range of f ( x ) = − ( 7 ) x and g ( x ) = 7 x is: f ( x ) and g ( x ) have the same domain but different ranges.
Examples
Understanding the domain and range of exponential functions is crucial in modeling real-world phenomena such as population growth and radioactive decay. For instance, if g ( x ) = 7 x represents the growth of a bacteria colony, the domain (all real numbers) indicates that we can theoretically consider the colony's size at any point in time. The range (positive real numbers) tells us that the colony's size will always be a positive value. Similarly, f ( x ) = − ( 7 ) x could represent the decay of a radioactive substance, where the negative sign indicates a decreasing quantity over time. The domain remains all real numbers, but the range (negative real numbers) reflects the decreasing nature of the substance.
The domain of both f ( x ) = − ( 7 ) x and g ( x ) = 7 x is all real numbers. However, the range of f ( x ) is all negative real numbers, while the range of g ( x ) is all positive real numbers. Therefore, the correct answer is that f ( x ) and g ( x ) have the same domain but different ranges.
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