Sure, let's analyze and compare the domain and range for the given functions:
### Function [tex]\( f(x) = 5x \)[/tex]
1. Domain:
- A linear function of the form [tex]\( f(x) = ax \)[/tex] typically has a domain of all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
- Thus, the domain of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
2. Range:
- The range of a linear function [tex]\( f(x) = 5x \)[/tex] can also take any real number value since [tex]\( x \)[/tex] can take any real number and multiplying by 5 will likewise produce any real number.
- Therefore, the range of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
### Function [tex]\( g(x) = 5^x \)[/tex]
1. Domain:
- An exponential function of the form [tex]\( g(x) = a^x \)[/tex], where [tex]\( a \)[/tex] is a positive constant, has a domain of all real numbers ([tex]\( -\infty < x < \infty \)[/tex]).
- So, the domain of [tex]\( g(x) = 5^x \)[/tex] is all real numbers.
2. Range:
- The range of an exponential function [tex]\( g(x) = 5^x \)[/tex] is all positive real numbers because for any real [tex]\( x \)[/tex], [tex]\( 5^x \)[/tex] produces a positive number. The function never reaches zero or a negative value.
- Therefore, the range of [tex]\( g(x) = 5^x \)[/tex] is [tex]\( y > 0 \)[/tex].
Based on these analyses, the correct options that Keshawn could include in his explanation are:
1. The domain of both functions is all real numbers.
2. The range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
These statements capture the correct properties for the domain and range of the given functions.
