Graphing a defined piecewise function

Question

F(x){ x² +1 if x<1

{2 if x=1

{7-5x if x>1

graph the function f.

Is this function continuous?

Answer 1

Hi Kelly,

To answer your question, Yes! The graph is continuous. Since we see that the first equation, f(x) = x² +1 continues in the x<1direction, and since we also know that parabolas are continuous, we know that we do not need to worry about whether or not the graph is continuous in the negative direction as Y approaches negative infinity. Similarly, we can see that the equation f(x) = 7-5x is a line that continues from, but not including, x=1 to infinity. We know that this line will continue to decrease at a slope of -5 units. However, since both of these equations do not include x = 1, we must make sure that x = 2 is where both the equations intersect in order for there to be a smooth line that we can draw continuously across (Tips and tricks like this can be learned in a tutor session!). To do this we have to calculate what f(x) is when x = 1 for both f(x) = x²+1 and f(x) = 7-5x.

f(x) = x² + 1 f(x) = 7 - 5x

f(1) = (1)² + 1 f(1) = 7 - 5(1)

f(1) = 1 + 1 f(1) = 7 - 5

f(1) = 2 and f(1) = 2

Since both equations intersect at the point (1,2) and the piecewise function let's us know that at x = 1, f(x) = 2, we know that the graph is continuous.

Good luck on any other questions you have and contact me if you need any other questions answered!