Video Transcript
Which of the following relations is satisfied by both the point negative one, one and the point zero, three. Is it (A) 𝑓 of 𝑥 is equal to four 𝑥 plus three, (B) 𝑓 of 𝑥 is equal to three 𝑥 plus three, (C) 𝑓 of 𝑥 is equal to two 𝑥 plus three, (D) 𝑓 of 𝑥 is equal to four 𝑥 plus five, or (E) 𝑓 of 𝑥 is equal to three 𝑥 plus four?
We begin by recalling that a relation is a function if for every element of the input set, there is an output and if no element of the input set is mapped to more than one element of the output. We know that functions can be represented as diagrams, in tables, in graphs, or as ordered pairs. In this question, we need to identify which of the five functions satisfies the two ordered pairs negative one, one and zero, three.
Any ordered pair is written in the form 𝑥, 𝑦, where the 𝑥-value is the input of our function and the 𝑦-value, the output. We will begin by substituting 𝑥 equals negative one into the five functions.
In option (A), 𝑓 of negative one is equal to four multiplied by negative one plus three. This is equal to negative four plus three, which equals negative one. As this is not equal to the corresponding 𝑦-coordinate of one, we can rule out option (A).
Repeating this for option (B), we have three multiplied by negative one plus three. This is equal to negative three plus three, which equals zero. So we can also rule out option (B).
For the function 𝑓 of 𝑥 is equal to two 𝑥 plus three, 𝑓 of negative one is equal to two multiplied by negative one plus three. This is equal to one, so the point negative one, one does satisfy the relation in option (C). This is also true for option (D). Four multiplied by negative one plus five is equal to one. Likewise, the point negative one, one satisfies the relation in option (E). 𝑓 of 𝑥 is equal to three 𝑥 plus four, since three multiplied by negative one plus four is equal to one.
We will now check the point zero, three by substituting 𝑥 equals zero into options (C), (D), and (E). In option (C), 𝑓 of zero is equal to two multiplied by zero plus three. This is equal to three. And therefore, the point also satisfies the relation 𝑓 of 𝑥 is equal to two 𝑥 plus three. This is not true for option (D), as four multiplied by zero plus five is equal to five. In the same way, for option (E), 𝑓 of zero is equal to three multiplied by zero plus four, which is equal to four.
We can therefore conclude that the relation that is satisfied by both the point negative one, one and the point zero, three is option (C) 𝑓 of 𝑥 is equal to two 𝑥 plus three.
