Consider teh funtion f whose graph is show below. What is the value of the follo

Question

Consider teh funtion f whose graph is show below. What is the value of the following?

3f(1)-2(limx->1+f(x))+5(limx->1-f(x))

Graph has a parabola that is open on -1,1 touches 0,0 and ends on 1,1. it then has a line at 1,2 with open end and ends at 3,0. I cannot figure out how to import the picture

Explanation / Answer

The important part here is to note that a function can only have one output for each input. We know that the line begins at (1,2) with open end, while the parabola ends at (1,1). Thus, at the input x=1, the function value is f(1)=1. Observe that the limit as x approaches 1 from the left side, or lim x->1-, is 1, while the limit as x approaches 1 from the right, or lim x->1+, is 2. Thus, 3f(1)-2(lim x->1+ f(x))+5(lim x->1- f(x))=3(1)-2(2)+5(1)=3-4+5=4. For the above to be 7, we need the parabola to end at (1,1) with open (unshaded) end and for the line to begin at (1,2) with closed (shaded) end. The one-sided limits would remain the same, but the function value for x=1 changes. f(1)=2. Thus, in this case, 3f(1)-2(lim x->1+ f(x))+5(lim x->1- f(x))=3(2)-2(2)+5(1)=6-4+5=7.

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