Question Video: Finding the Equation That Represents Given Mapping Diagrams Mathematics • Third Year of Preparatory School

Video Transcript

Given that 𝑅 is a relation from 𝑥 to 𝑦, where 𝑎 exists in 𝑥 and 𝑏 exists in 𝑦, which of the following equations correctly expresses relation 𝑅? Is it (A) 𝑏 equals 𝑎 plus one, (B) 𝑏 equals two 𝑎 plus two, (C) 𝑏 equals two 𝑎 minus two, (D) 𝑎 is equal to two 𝑏 plus two, or (E) 𝑎 is equal to two 𝑏 minus two?

Any relation 𝑅 contains a set of ordered pairs of the form 𝑥, 𝑦. In the diagram shown, we have three ordered pairs: negative one, zero; four, 10; and five, 12. We can therefore say that the relation 𝑅 is the set of these three ordered pairs. We are asked to find the correct equation that matches any value in 𝑥 𝑎 to a value in 𝑦 𝑏. The easiest way to do this is to substitute our values into each of the equations. Let’s begin with the ordered pair negative one, zero.

We will let 𝑎 equal negative one and 𝑏 equal zero. Zero is equal to negative one plus one. This means that equation (A) does work for the first ordered pair. Likewise, two multiplied by negative one plus two is also equal to zero. This means that equation (B) also works for the first ordered pair. In option (C), two multiplied by negative one minus two is equal to negative four and not zero. This means that equation (C) is not the correct answer. This is also true of options (D) and (E) as negative one is not equal to two multiplied by zero plus two or two multiplied by zero minus two. We can therefore rule out both of these options.

We will now consider the second ordered pair four, 10 for equation (A) and equation (B). This time, 𝑎 is equal to four and 𝑏 is equal to 10. 10 is not equal to four plus one. This means that equation (A) is also incorrect. 10 is equal to two multiplied by four plus two. This means that equation (B) holds for the first and second ordered pairs. We can now move on to the third ordered pair five, 12. Two multiplied by five plus two is equal to 12. As equation (B) holds for all three ordered pairs, this is the correct answer. The equation that correctly expresses relation 𝑅 is 𝑏 is equal to two 𝑎 plus two.