To determine what the variable [tex]\( x \)[/tex] represents in the equation [tex]\( f(x) = 0.5 + 2x \)[/tex], we need to carefully analyze the given context and the structure of the equation itself.
1. Understanding the Initial Condition:
- On his first run, Sean runs [tex]\(0.5\)[/tex] miles. This initial condition is represented by the constant term [tex]\(0.5\)[/tex] in the equation.
2. Understanding the Increment:
- Sean increases his workout by adding [tex]\(2\)[/tex] miles every month. This indicates that for each month that passes, an additional [tex]\(2\)[/tex] miles are added to his initial [tex]\(0.5\)[/tex] miles. This increment is represented by the coefficient [tex]\(2\)[/tex] times the variable [tex]\(x\)[/tex].
3. Structure of the Equation:
- The equation is [tex]\( f(x) = 0.5 + 2x \)[/tex], where [tex]\(f(x)\)[/tex] represents the total number of miles Sean runs after a certain number of months.
4. Analyzing the Variable [tex]\( x \)[/tex]:
- The variable [tex]\(x\)[/tex] scales the increment. Considering he adds [tex]\(2\)[/tex] miles every month, [tex]\(x\)[/tex] must represent the number of months he has been running. For example, if [tex]\(x = 1\)[/tex], then he has been running for 1 month and the distance he runs becomes [tex]\(f(1) = 0.5 + 2 \times 1 = 2.5\)[/tex] miles; for [tex]\(x = 2\)[/tex], it becomes [tex]\(f(2) = 0.5 + 2 \times 2 = 4.5\)[/tex] miles, and so on.
Based on this analysis, the variable [tex]\(x\)[/tex] in the equation [tex]\( f(x) = 0.5 + 2x \)[/tex] represents the number of months he runs.
Thus, the correct answer is:
months he runs.
