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Could someone help me in this problem: I have a nondecreasing function $f(x)$ on $\mathbb R$ (so we know how the graph will looks like). How can we graph the functions $ax$, $f(x)$, and $ax+f(x)$, (a>0) all on the same $xy-$coordinate; which one is above the other?

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    I'm interested in $a=4$.2012-01-28

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Here's an example:

enter image description here

Note that

  • The graph of $y=ax$ is a straight line passing through the origin with positive slope (for $a>0$).
  • The graph of $g(x)=f(x)+ax$ can be obtained by "adding heights (or depths)" as indicated in the diagram.
  • For $x>0$, the graph of $g(x)=f(x)+ax$ is above the graph of $y=f(x)$; because, in this case, to find the value of $g(x)$, you add a positive value, that of $ax$, to $f(x)$.
  • For $x<0$, the graph of $g(x)=f(x)+ax$ is below the graph of $y=f(x)$; because, in this case, to find the value of $g(x)$, you subtract a positive value, that of $|ax|$, from $f(x)$.
  • For $x=0$, we have $g(0)=f(0)+0=f(0)$.