Determine 𝑓 evaluated at negative two.
In this question, we’re given the graph of a piecewise-defined function. And we need to use this graph to determine the value of 𝑓 evaluated at negative two. And to do this, we recall the 𝑥-coordinate of any point on our graph will be the input value of 𝑥 and the corresponding 𝑦-coordinate will be the output of our function. For example, because our graph goes through the point with coordinates two, four, we can conclude that 𝑓 evaluated at two is equal to four. We want to evaluate 𝑓 at negative two. So our input value of 𝑥 is negative two.
One way of doing this is to sketch a vertical line 𝑥 is equal to negative two and determine the point of intersection with our graph. And we can see there are two possible points of intersection between the vertical line and the graph of our piecewise function. There’s the solid dot at the point negative two, zero, and there’s the hollow dot at the point negative two, two. However, we can recall when there’s a hollow dot on our graph, that means our function is not defined at this point and when there’s a solid dot, our function does pass through this point. So, the graph of our function passes through the point negative two, zero. And therefore, 𝑓 evaluated at negative two is equal to zero.
Therefore, by considering the points of intersection between the given graph and the vertical line 𝑥 is equal to negative two, we were able to show that 𝑓 evaluated at negative two is equal to zero.