To determine if a relationship describes a function, we need to see if each input (independent variable) corresponds to exactly one output (dependent variable).
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Is the length of a television show a function of the age of the director?
- In this case, the "age of the director" is the input, and the "length of a television show" is the output. Each age could correspond to different shows of varying lengths, meaning that one age of the director could be associated with multiple show lengths. Thus, this relationship does not define a function because a single input (age of director) could produce multiple outputs (lengths of different TV shows).
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Is the amount of juice in cups a function of the amount of juice in ounces?
- Here, the input is "number of ounces" and the output is "number of cups." Since there are 8 ounces in a cup, you can determine the number of cups from the number of ounces using the formula: \[ \text{Number of cups} = \frac{\text{Number of ounces}}{8} \] This means that for every specific amount of ounces, there is exactly one corresponding amount of cups. Therefore, this relationship does describe a function.
To summarize:
- The length of a television show is not a function of the age of the director.
- The amount of juice in cups is a function of the amount of juice in ounces.
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