Keshawn is asked to compare and contrast the domain and range for the two functions.
[tex]
\begin{array}{l}
f(x)=5 x \\
g(x)=5^x
\end{array}
[/tex]
Which statements could he include in his explanation? Select two options.
A. The domain of both functions is all real numbers.
B. The domain of [tex]f(x)[/tex] is [tex]x\ \textgreater \ 5[/tex].
C. The domain of [tex]g(x)[/tex] is [tex]x\ \textgreater \ 5[/tex].
D. The range of [tex]f(x)[/tex] is [tex]y\ \textgreater \ 0[/tex].
E. The range of [tex]g(x)[/tex] is [tex]y\ \textgreater \ 0[/tex]
Answer (1)
The domain of f ( x ) = 5 x is all real numbers.
The range of f ( x ) = 5 x is all real numbers.
The domain of g ( x ) = 5 x is all real numbers.
The range of g ( x ) = 5 x is 0"> y > 0 .
The two correct statements are: The domain of both functions is all real numbers, and the range of g ( x ) is 0"> y > 0 .
0.}"> The domain of both functions is all real numbers. The range of g ( x ) is y > 0.
Explanation
Understanding the Problem We are given two functions, f ( x ) = 5 x and g ( x ) = 5 x , and we need to compare their domains and ranges to select two correct statements from the given options.
Domain of f(x) Let's analyze the domain of f ( x ) = 5 x . Since f ( x ) is a linear function, it is defined for all real numbers. Therefore, the domain of f ( x ) is all real numbers.
Range of f(x) Now let's analyze the range of f ( x ) = 5 x . Since f ( x ) is a linear function, it can take any real value. Therefore, the range of f ( x ) is all real numbers.
Domain of g(x) Let's analyze the domain of g ( x ) = 5 x . Since g ( x ) is an exponential function, it is defined for all real numbers. Therefore, the domain of g ( x ) is all real numbers.
Range of g(x) Now let's analyze the range of g ( x ) = 5 x . Since g ( x ) is an exponential function with a positive base (5), its output is always positive. Therefore, the range of g ( x ) is 0"> y > 0 .
Selecting Correct Statements Now, let's compare our findings with the given options:
The domain of both functions is all real numbers. - This is correct.
The domain of f ( x ) is 5"> x > 5 . - This is incorrect.
The domain of g ( x ) is 5"> x > 5 . - This is incorrect.
The range of f ( x ) is 0"> y > 0 . - This is incorrect.
The range of g ( x ) is 0"> y > 0 . - This is correct.
Therefore, the two correct statements are:
The domain of both functions is all real numbers.
The range of g ( x ) is 0"> y > 0 .
Examples
Understanding the domains and ranges of functions is crucial in many real-world applications. For example, when modeling population growth with an exponential function like g ( x ) = 5 x , knowing that the range is 0"> y > 0 tells us that the population will always be a positive number. Similarly, in linear models like f ( x ) = 5 x , understanding that the domain is all real numbers allows us to input any value for x , such as time or resources, without mathematical restrictions. These concepts are fundamental in fields like biology, economics, and engineering, where mathematical models are used to predict and analyze various phenomena.
