To determine the correct piecewise function from the options provided, we need to analyze the ranges given for \( x \) (the number of hours a bicycle is rented) and the corresponding values of \( y \) (the cost in dollars).
- The first segment \( 8x \) typically suggests that the cost is \( 8 \) dollars per hour for \( 0 \leq x < 3 \).
- The next segment \( 7x \) implies a rate of \( 7 \) dollars per hour from \( 3 \) to \( 5 \) hours.
- The last segment \( 6x \) indicates a rate of \( 6 \) dollars per hour for \( x \geq 5 \).
Additionally, we check for equalities/inequalities at the boundary points to see if they are included (closed) or excluded (open).
From the options provided:
- Option #1 suggests ranges where the lower boundary of the second segment is an exclusive value and the first boundary of the third segment is also an exclusive value, which may not accurately reflect certain conditions.
- Option #2 suggests including both endpoints for the second segment including \( x=3 \) and excluding \( x=5 \), which might not fit if the function starts with \( 7x \) at exactly \( 3 \) and should be \( 6x \) starting at \( 5 \).
- Option #3 includes \( 3 \) as included and \( 5 \) as included when transitioning from one segment to the next, which seems more logical since \( 7x \) should be valid at \( 3 \) and becomes \( 6x \) immediately after.
Hence, the correct answer seems to be Option #3: \[ y = \begin{cases} 8x, & 0 \leq x < 3 \ 7x, & 3 \leq x < 5 \ 6x, & x \geq 5 \end{cases} \]
This accurately reflects a piecewise function where the transition from one rate to another occurs at the specified bounds.
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