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Worked example: two inputs with the same output (graph)

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The graph of the function f is shown below. What is the input value other than negative five for which f of x is equal to f of negative five? So, we have our x axis, we have our y axis. And then in blue, they've graphed y equals f of x. So for example when x is equal to one, f of x, when y is going to be equal to f of x --that's what this graph is-- f of x is equal to one. When x is equal to seven, f of seven, we see, is equal to five. When x is equal to nine, we see that f of nine is equal to six. So what is the input value other than negative five for which f of x is equal to f of negative five? So, let's see. If x is equal to negative five, f of negative five is... -- we move up here to the graph-- It is equal to four. Cause this, once again, is the graph y is equal to f of x. So the y coordinate here, this is what f of x, this is what five of negative five is equal to. So f of negative five is equal to four. So where else does that happen? So let's see. Let's just move horizontally to the right. That happens here as well. So what input do we have to give? What's the x coordinate here to get y is equal to f of x or f of x is equal to four right over here? Well, we see the x coordinate is also four. So this tells us that f of four is equal to four, which is the same thing as f of negative five. So when x is four, the function takes on the same value as when x is negative five. So x equals four. And we are done.