# Question Video: Finding the Range of a Linear Function Given That It Maps from a Given Interval to Real Numbers Mathematics

If 𝑓∶ [2, 21] → ℝ, where 𝑓(𝑥) = 3𝑥 − 10, find the range of 𝑓.

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### Video Transcript

If 𝑓 maps elements on the closed interval from two to 21 to the set of real numbers, where 𝑓 of 𝑥 is equal to three 𝑥 minus 10, find the range of 𝑓.

In this question, we are given a linear function 𝑓 of 𝑥, which is equal to three 𝑥 minus 10. We know that the graph of 𝑦 is equal to three 𝑥 minus 10 is a straight line as shown. We know that the domain of a function is the set of input or 𝑥-values. And in this question, we’re told that these lie on the closed interval from two to 21. The function 𝑓 of 𝑥 is therefore defined for all values between the two points shown.

We can calculate the corresponding 𝑓 of 𝑥 values by substituting 𝑥 equals two and 𝑥 equals 21 into the function. When 𝑥 is equal to two, 𝑓 of 𝑥 is equal to three multiplied by two minus 10. This is equal to negative four. Likewise, when 𝑥 is equal to 21, 𝑓 of 𝑥 is equal to three multiplied by 21 minus 10. This is equal to 53. Since the range of a function 𝑓 is the set of outputs or 𝑦-values, we can conclude that 𝑓 of 𝑥 or 𝑦 is greater than or equal to negative four and less than or equal to 53. Using interval notation, the range of the function 𝑓 on the given domain is the closed interval from negative four to 53.

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