Composite of piecewise functions

[smithnya]

Homework Statement

Find f°g for each pair of functions f and g.

Homework Equations

f(x) = {x+1, if x≤0 or 2x, if x>0

g(x) = {2x, if x≤-1 or -x, if x>-1

The Attempt at a Solution

I am able to determine the following:

(f°g)(x) = {2x+1, -2x, -x+1

However, these functions carry intervals, and I don't know how to obtain those intervals

For example, my book gives the following solution,

(f°g)(x) = {2x+1, if x≤-1 or -2x, if -1<x<0 or -x+1, if x≥0

I can obtain the equations, but I don't know how to determine the actual intervals for the individual equations.

Could someone please explain it to me?

 

[SammyS]

Homework Statement

Find f°g for each pair of functions f and g.

Homework Equations

f(x) = {x+1, if x≤0 or 2x, if x>0

g(x) = {2x, if x≤-1 or -x, if x>-1

The Attempt at a Solution

I am able to determine the following:

(f°g)(x) = {2x+1, -2x, -x+1

However, these functions carry intervals, and I don't know how to obtain those intervals

For example, my book gives the following solution,

(f°g)(x) = {2x+1, if x≤-1 or -2x, if -1<x<0 or -x+1, if x≥0

I can obtain the equations, but I don't know how to determine the actual intervals for the individual equations.

Could someone please explain it to me?

It may help to graph the functions.

You're looking for (f○g)(x) = f(g(x)) .

You will have to look at g(x), to see what portion of the domain of g(x) has a range that's in the interval, (-∞, 0] and what portion of the domain of g(x) has a range that's in the interval, (0, +∞) .