Which of the following represents a function with input 𝑥 and output 𝑦?
First, let’s define function. A function relates an input to an output such that every input in the function is related to only one output. Let’s try to visualize relation A and relation B in a different way. If the input values are on the left and the output values are on the right, then our 𝑥-values will go on the left. And our 𝑦-values will go on the right. For relationship A, we’ll have negative three, zero, three, eight, and negative 10 in the input. And we’ll have six, eight, 20, and four in the outputs. Notice that there’re only five values in the output. That’s because there are two values of eight in the outputs. And so we only need to put an eight once.
And now, we’ll connect our input to our output values. Negative three to six, zero to eight, three to 20, eight to four, and negative 10 to eight. We’ll do the same thing for relationship B. Notice that, in the input, we only list negative two once since we have the input value negative two twice in the table. The output values are the same as the output from relationship A. So we’ll list six, eight, 20, and four. Again, we only list eight once. Now, we need to connect the relationships, negative two to six and negative two to 20. Negative two has two output values, zero to eight, seven to four, and negative eight to eight.
But we know that, in a function, every input can only be related to one output. But in relationship B, negative two, the input value, is connected to six and to 20 for the output value, which means that relationship B cannot be a function. You might notice in relationship A the output value of eight has two lines connected to it. And that’s okay. Multiple 𝑥-values can be connected to the same 𝑦-value. But multiple 𝑦-values cannot be connected to the same 𝑥-value.
And so we say that relation A is the function.