Question
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  1. Higher Order Thinking Consider the functions
    and for the domain
    .
    a. Without evaluating or graphing the
    functions, how do the ranges compare?

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Answer

The range of is and the range of is . The range of is entirely below the range of .

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Bonus Knowledge

Both functions and are linear, with having a slope of 2 and a y-intercept of 1, while has the same slope but a y-intercept of 2. Since both have the same slope, they will be parallel lines on a graph. Over the domain , will produce values ranging from 1 to 11, and will produce values ranging from 2 to 12. Therefore, the range of will always be shifted up by 1 unit compared to the range of .
In summary, while the ranges of both functions overlap in their respective spans, 's range will consistently be 1 unit higher for all corresponding values of in the same domain.

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