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

If π‘“βˆΆ [2, 21] β†’ ℝ, where 𝑓(π‘₯) = 3π‘₯ βˆ’ 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 π‘₯ is equal to three π‘₯ minus 10, find the range of 𝑓.

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

We can calculate the corresponding 𝑓 of π‘₯ values by substituting π‘₯ equals two and π‘₯ equals 21 into the function. When π‘₯ is equal to two, 𝑓 of π‘₯ is equal to three multiplied by two minus 10. This is equal to negative four. Likewise, when π‘₯ is equal to 21, 𝑓 of π‘₯ 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 π‘₯ 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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