What is the domain and range for # f(x) = 3x - absx#?

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Answer 1 · 2015-10-14T16:56:58

Both the domain and the range are the whole of #RR#.

#f(x) = 3x-abs(x)# is well defined for any #x in RR#, so the domain of #f(x)# is #RR#.

If #x >= 0# then #abs(x) = x#, so #f(x) = 3x-x = 2x#.

As a result #f(x)->+oo# as #x->+oo#

If #x < 0# then #abs(x) = -x#, so #f(x) = 3x + x = 4x#.

As a result #f(x)->-oo# as #x->-oo#

Both #3x# and #abs(x)# are continuous, so their difference #f(x)# is continuous too.

So by the intermediate value theorem, #f(x)# takes all values between #-oo# and #+oo#.

We can define an inverse function for #f(x)# as follows:

#f^(-1)(y) = { (y/2, "if " y >= 0), (y/4, "if " y < 0) :}#

graph{3x-abs(x) [-5.55, 5.55, -2.774, 2.774]}