Which statement best describes the domain and range of [tex]p(x)=6^{-x}[/tex] and [tex]q(x)=6^x[/tex] ?
A. [tex]p(x)[/tex] and [tex]q(x)[/tex] have the same domain and the same range.
B. [tex]p(x)[/tex] and [tex]q(x)[/tex] have the same domain but different ranges.
C. [tex]p(x)[/tex] and [tex]q(x)[/tex] have different domains but the same range.
D. [tex]p(x)[/tex] and [tex]q(x)[/tex] have different domains and different ranges.
Answer (1)
Determine the domain of p ( x ) = 6 − x , which is all real numbers.
Determine the range of p ( x ) = 6 − x , which is all positive real numbers.
Determine the domain of q ( x ) = 6 x , which is all real numbers.
Determine the range of q ( x ) = 6 x , which is all positive real numbers. Therefore, p ( x ) and q ( x ) have the same domain and the same range: p ( x ) and q ( x ) have the same domain and the same range .
Explanation
Understanding the Problem We are given two exponential functions, p ( x ) = 6 − x and q ( x ) = 6 x . Our goal is to determine and compare their domains and ranges. The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.
Analyzing p(x) For the function p ( x ) = 6 − x , we can input any real number for x . There are no restrictions on the values of x . Therefore, the domain of p ( x ) is all real numbers. Since 6 − x is equivalent to ( 6 1 ) x , and any positive number raised to any real power is always positive, the range of p ( x ) is all positive real numbers.
Analyzing q(x) For the function q ( x ) = 6 x , similar to p ( x ) , we can input any real number for x . There are no restrictions on the values of x . Therefore, the domain of q ( x ) is all real numbers. Since 6 x is an exponential function with a positive base, its output is always positive. Therefore, the range of q ( x ) is all positive real numbers.
Comparing Domains and Ranges Comparing the domains and ranges of p ( x ) and q ( x ) , we find that both functions have the same domain (all real numbers) and the same range (all positive real numbers). Therefore, the statement that best describes the domain and range of p ( x ) and q ( x ) is: p ( x ) and q ( x ) have the same domain and the same range.
Final Answer Therefore, the correct answer is that p ( x ) and q ( x ) have the same domain and the same range.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding the domain and range of these functions is crucial for making accurate predictions and interpreting the results. For example, if we are modeling the decay of a radioactive substance using an exponential function, the domain would represent the time elapsed since the start of the decay, and the range would represent the amount of the substance remaining at that time. Knowing that the range is always positive helps us understand that the substance will never completely disappear, but will only approach zero asymptotically.
