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)
The domain of p ( x ) = 6 − x is all real numbers.
The range of p ( x ) = 6 − x is all positive real numbers.
The domain of q ( x ) = 6 x is all real numbers.
The range of q ( x ) = 6 x 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 functions, p ( x ) = 6 − x and q ( x ) = 6 x . We need to determine their domains and ranges to see which statement is correct.
Domain of p(x) Let's find the domain of p ( x ) = 6 − x . Since we can plug in any real number for x , the domain is all real numbers, or ( − ∞ , ∞ ) .
Range of p(x) Now let's find the range of p ( x ) = 6 − x . Since 6 − x is always positive for any real number x , the range is all positive real numbers, or ( 0 , ∞ ) .
Domain of q(x) Let's find the domain of q ( x ) = 6 x . Since we can plug in any real number for x , the domain is all real numbers, or ( − ∞ , ∞ ) .
Range of q(x) Now let's find the range of q ( x ) = 6 x . Since 6 x is always positive for any real number x , the range is all positive real numbers, or ( 0 , ∞ ) .
Comparison Comparing the domains and ranges, we see that both functions have the same domain (all real numbers) and the same range (all positive real numbers).
Conclusion Therefore, the statement that best describes the domain and range of p ( x ) = 6 − x and q ( x ) = 6 x is: p ( x ) and q ( x ) have the same domain and the same range.
Examples
Understanding the domain and range of exponential functions is crucial in various real-world applications. For instance, in modeling population growth or radioactive decay, the domain represents time (which can be any real number), and the range represents the population size or the amount of radioactive material (which is always positive). Knowing the domain and range helps us interpret the model's results and make accurate predictions.
