Question Video: Evaluating Algebraic Expressions given a Function on a Given Set in Listing Method Mathematics • 8th Grade

Given that 𝑋 = {βˆ’7, βˆ’1, 9}, and 𝑅 {(π‘Ž, βˆ’1), (𝑏, βˆ’7), (βˆ’7, 9)}, where 𝑅 is a function on 𝑋, find the numerical value of π‘Ž + 𝑏.

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Video Transcript

Given that 𝑋 equals the set containing negative seven, negative one, nine and 𝑅 is the set containing the ordered pairs π‘Ž, negative one; 𝑏, negative seven; negative seven, nine, where 𝑅 is a function on 𝑋, find the numerical value of π‘Ž plus 𝑏.

Let’s begin by looking at some of the notation and terminology in the question. We’re told that 𝑅 is a function on 𝑋. This means that 𝑅 takes values in the set 𝑋 and maps them onto exactly one value in a second set. The values in 𝑋 are the input to the function. So we say the 𝑋 is its domain. Let’s define then the range of the function to be some second set π‘Œ. This is the set of possible outputs to the function. And in fact, we’re told that 𝑋 contains the elements negative seven, negative one, nine. So, what does π‘Œ contain?

Well, the ordered pairs can help us. The first value in each ordered pair is the input. So we choose that from set 𝑋. Then, the second value is the output. So we choose that from set π‘Œ. This means set π‘Œ must contain the values negative one, negative seven, nine. And in fact, we can identify one of the mappings straightaway. The third ordered pair tells us that negative seven is mapped onto nine through function 𝑅. So what does this mean for our other two ordered pairs?

Consider the first one: π‘Ž, negative one. We know that exactly one number in set 𝑋 will map onto the number negative one in set π‘Œ. Similarly, exactly one number, 𝑏, in set 𝑋 will map onto the number negative seven. For the sake of this question, it actually doesn’t matter if we define π‘Ž to be negative one and 𝑏 to be nine or the other way round. In the first case, if we let π‘Ž be equal to negative one and 𝑏 be equal to nine, our first two ordered pairs become negative one, negative one and nine, negative seven.

This is equivalent to the mapping diagram we’ve drawn. But we could, of course, map nine onto negative one and negative one onto negative seven. We don’t know which one is which, but we do know that, in either case, when we add π‘Ž and 𝑏, we are going to be adding negative one and nine. Remember, addition is commutative, so this can be done in any order. So negative one plus nine is the same as nine plus negative one. It’s eight. And so we found a numerical value of π‘Ž plus 𝑏; it’s eight.

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