To determine the range of the function [tex]\( m(x) = 30x + 5,000 \)[/tex], let's analyze the function step by step.
1. Understanding the Function:
- The given function [tex]\( m(x) = 30x + 5,000 \)[/tex] represents the total mileage on the car.
- Here, [tex]\( m(x) \)[/tex] is the total mileage and [tex]\( x \)[/tex] is the number of gallons of gas consumed.
2. Interpretation of the Function:
- When [tex]\( x = 0 \)[/tex], the initial mileage of the car is given by [tex]\( m(0) \)[/tex]:
[tex]\[
m(0) = 30 \cdot 0 + 5,000 = 5,000
\][/tex]
So, the car starts with 5,000 miles.
- As [tex]\( x \)[/tex] increases (i.e., more gas is consumed), the mileage [tex]\( m(x) \)[/tex] will also increase because [tex]\( 30x \)[/tex] is a positive term.
3. Identifying the Range:
- Because the function [tex]\( m(x) = 30x + 5,000 \)[/tex] is linear with a positive slope (30), it means that as [tex]\( x \)[/tex] increases from 0 to positive infinity, [tex]\( m(x) \)[/tex] will increase from 5,000 to positive infinity.
- The lowest value of [tex]\( m(x) \)[/tex] occurs at [tex]\( x = 0 \)[/tex], which is 5,000.
Thus, the set of all possible values of [tex]\( m(x) \)[/tex] starts from 5,000 and increases indefinitely. This means that the range of the function can be expressed as:
[tex]\[
[5,000, \infty)
\][/tex]
Therefore, the correct answer is:
[tex]\[ [5,000, \infty) \][/tex]
