SOLUTION: Suppose that the function f is defined, for all real numbers, as follows. f(x)={ x-2 if x<-2 { 3x+2 if x>=-2 Graph the function f. Then determine whether or not the fun

Question 1188881: Suppose that the function f is defined, for all real numbers, as follows.
f(x)={ x-2 if x<-2
{ 3x+2 if x>=-2
Graph the function f. Then determine whether or not the function is continuous.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

The graph is here

I used GeoGebra to make the graph.

We graph the two equations y = x-2 and y = 3x+2; however, we only graph pieces of each. The first equation is only graphed when x < -2. The second piece is graphed when x = -2 or larger.

The two pieces are connected, so therefore, the entire piecewise function is continuous.

We can prove that this function is continuous by noting that the pieces themselves are linear, so they are naturally continous on their own. The only thing we need to check is the junction point (if there is one). Let's plug in x = -2 into each piece. If we get the same y value each time, then this is sufficient to prove continuity.

y = x-2
y = -2-2
y = -4
and
y = 3x+2
y = 3(-2)+2
y = -6+2
y = -4
We get the same y output for x = -2, so the two pieces connect together. This algebraically backs up what the graph above shows.

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