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Question Video: Finding the Domain and Range of an Absolute Value Function Mathematics • 10th Grade

Determine the domain and the range of the function 𝑓(𝑥) = |−𝑥 − 1| + 1.


Video Transcript

Determine the domain and range of the function 𝑓 of 𝑥 equals the absolute value of negative 𝑥 minus one plus one.

The domain represents the 𝑥-values that the graph covers and the range represents the 𝑦-values that the graph covers. So we will look at the 𝑥-axis for the domain and the 𝑦-axis for the range. So looking at our 𝑥-axis, let’s begin at zero.

At zero, we can land at two. At one, we can land at three. Two, we land at four. Three, we land at five. And at every integer, we can actually see that the graph has a place where this number can go. And actually in between all of those integers, any decimal between those also has a place to go. And notice our graph has arrows at the end, so this keeps extending forever left and right. So this means our domain is all real numbers. So the domain would represent all the real numbers.

So looking at the range now, beginning at zero, there’s nowhere where 𝑦 equals zero for this graph to go. And going down into the negative numbers, again we’re not getting any closer to our graph and there’s nowhere for it to land. So, so far, the range doesn’t have any 𝑦-values from the values that we looked at. So let’s go up towards the positive values.

Around one half, there’s still nowhere to land. But finally at 𝑦 equals one, we can finally start landing somewhere. And as we go up in the positive direction for 𝑦, there’s somewhere to land. So starting with one and increasing up into the positive numbers, we have a place to go. So the range is from one to infinity with a bracket on the one, because we can actually land there. If we couldn’t land on one, then we would use a parenthesis. So again the domain is all real numbers and the range is from one to infinity.

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