Question

keshawn is asked to compare and contrast the domain and range for the two functions.
$f(x) = 5x$
$g(x) = 5^x$
which statements could he include in his explanation?
select two options.

• the domain of both functions is all real numbers.
• the domain of $f(x)$ is $x > 5$.
• the domain of $g(x)$ is $x > 5$.
• the range of $f(x)$ is $y > 0$.
• the range of $g(x)$ is $y > 0$.

Explanation:

1. For the function \( f(x) = 5x \), it is a linear function. The domain of a linear function (a polynomial of degree 1) is all real numbers because we can plug in any real number for \( x \) and get a valid output. The range is also all real numbers since as \( x \) takes all real values, \( 5x \) will also take all real values.
2. For the function \( g(x)=5^{x} \), it is an exponential function. The domain of an exponential function of the form \( a^{x} \) (where \( a>0,a

eq1 \)) is all real numbers because we can raise 5 to any real power. The range of \( g(x) = 5^{x} \) is \( y>0 \) because any positive number raised to a real power is positive, and it can get arbitrarily close to 0 (as \( x
ightarrow-\infty \)) but never actually be 0, and it can grow without bound (as \( x
ightarrow\infty \)).

Now let's analyze each option:

• "The domain of both functions is all real numbers." - This is correct because the domain of \( f(x)=5x \) (linear) is all real numbers and the domain of \( g(x) = 5^{x} \) (exponential) is also all real numbers.
• "The domain of \( f(x) \) is \( x > 5 \)." - This is incorrect. The domain of \( f(x)=5x \) is all real numbers, not restricted to \( x>5 \).
• "The domain of \( g(x) \) is \( x > 5 \)." - This is incorrect. The domain of \( g(x)=5^{x} \) is all real numbers, not restricted to \( x > 5 \).
• "The range of \( f(x) \) is \( y > 0 \)." - This is incorrect. The range of \( f(x)=5x \) is all real numbers, not just \( y>0 \).
• "The range of \( g(x) \) is \( y > 0 \)." - This is correct, as explained above for the exponential function \( g(x)=5^{x} \).

Answer:

• The domain of both functions is all real numbers.
• The range of \( g(x) \) is \( y > 0 \).