Video: Finding the Range of Piecewise-Defined Functions

Determine the range of the function. 𝑓(π‘₯) = π‘₯ βˆ’ 1 if π‘₯ ∈ [1, 6] and 𝑓(π‘₯) = βˆ’5π‘₯ + 35 if π‘₯ ∈ ]6, 7]

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Video Transcript

Determine the range of the function.

Remember that the range of the function is the output, the function of π‘₯, or the 𝑦 value. This data about this set of π‘₯ values is called the domain. We’ll need to use the domain to determine the range.

Our domain can be from one to seven. We’ll make a table to determine what the range is going to be. Our π‘₯ values begin at one and end at seven.

When π‘₯ is equal to one, we use the function π‘₯ minus one. One minus one is zero. When π‘₯ is two, we’re still dealing with our first function. Two minus one equals one. Three falls in the first function. Three minus one is two. Four is still the first function. Four minus one equals three. Five is the first function. Five minus one equals four. We look closely at six, but we see that the bracket is facing inward. So it is contained in the first function. Six minus one equals five.

And last, we have the number seven. If our π‘₯ value is seven, we use the function negative five times π‘₯ plus 35. The function of seven equals negative five times seven plus 35: negative 35 plus 35. When π‘₯ equals seven, the output is zero. Our range comes from our 𝑓 of π‘₯ values, the smallest of these being zero, and working all the way up until five. The range of the function is from zero to five.

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