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Modulus function graph question

Hello, I've never encountered this sort of question before and I don't know how to approach it. Could someone explain it please?

The functions f and g are defined on the domain of all real numbers by f(x)= |x-2| and g(x)= |x|-2.

Sketch the graph of f(x) - g(x).

Reply 1

davros

Original post by Engineering Lad

OK so you know that f(x) - g(x) = |x-2| - |x| + 2 and you need to work out how this behaves in different regions. The critical points are at x = 2 and x = 0 because these are where x-2 and x change sign, so you just need to sort out the behaviour in the regions
x < 0
0 < x < 2
x > 2

Reply 2

WhiteGroupMaths

First, let us define each of the individual modulus functions:

|x-2| = x-2 if x≥2
= 2-x if x<2

|x|= x if x≥0
= -x if x<0

There are 3 critical regions, namely x<0, 0 ≤ x < 2 and x≥2

For the extreme left critical region, ie x<0,

f(x) - g(x) = |x-2| - |x| + 2 = (2-x) - (-x) +2 = 4

In other words, you shall draw a horizontal line y=4 all the way from x=-∞ to x=0.

Thereafter,

for the next critical region 0 ≤ x < 2,

f(x) - g(x) = |x-2| - |x| + 2 = (2-x) - (x) +2 = 4-x

In this case, you shall draw the line y=4-x from x=0 to x=2.

I shall let you figure out the final graph you need to draw for the remaining critical region, which shouldn't be all too difficult if you can understand what I have explained thus far.

Hope it helps. Peace.

(edited 9 years ago)

Reply 3

RoyalBlue7

Original post by WhiteGroupMaths

Thereafter,

for the next critical region 0 ≤ x < 2,

f(x) - g(x) = |x-2| - |x| + 2 = (2-x) - (x) +2 = 4- *x

In this case, you shall draw the line y=4- *x from x=0 to x=2.

It should be y=4 - 2x, shouldn't it?