Continuity

Take out assignment from Tuesday

Continuous

• The graph of a piecewise function is said to be continuous if you can trace the graph with your finger without lifting your finger off the page.

• When there are an infinite number of points on the graph that are always connected.

Even though the graph looks like a parabola, a line with a positive slope, a line with zero slope and an exponential function all in one, this piecewise function is said to be continuous because it does not

break from one part of the function to the next.

The piecewise function shown in this graph is continuous because there are no "gaps" or "breaks" in the plotting

In this example, the domain is All Reals since all x-values have a plotted value

Notice that the "changes" focus around the x-values of 1 and -1.

♦ Hint: When graphing, focus on where the changes in the graph occur.

From x-values of -∞ to -1, the graph is a straight line.

From x-values of -1 to 1, the graph is constant.

From x-values 1 to ∞, the graph is quadratic (part of a parabola).

Warm-upWhat does b have to be to make this graph continuous?

1. f(x) = -x + b x ≤ - 2 -x – 3 x > -2

2. f(x) = -3x²- b , x ≤ -1 5x – 1 , x > -1

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What does b have to be to make this graph continuous?

1. F(x) = 2x + b x ≤ - 2 -2x – 2 x > -2{

Answer: 2x + b = -2x -2 at x = -2 2(-2) + b = -2(-2) – 2 -4 + b = 4 – 2 -4 + b = 2 +4 +4 b = 6