Problems from Graphic determination of the domain and of the image

Consider the following function defined in parts:

f(x)={x+2six02si0<x2x+4six>2

Do the graphic representation. Find the domain and the image of the function.

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Development:

A way of proceeding is to draw the graph of the function and then find the domain and the image.

We may realize that:

  • In the interval (,0] we have a straight line of slope m=1 and that cuts the axis x in x=2.

  • In the interval (0,2], we have a constant function y=2.

  • In the interval (2,+) we have a straight line of slope m=1 and that cuts the axis x in x=4.

Therefore the graph of the function is:

imagen

Thus it is clear that the domain of the function is:

Dom(f)=(,+)

and that its image is:

Im(f)=(,2]

Solution:

Dom(f)=(,+)

Im(f)=(,2]

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