# Thread: problem with sketching a graph of composite function

1. ## problem with sketching a graph of composite function

graph the function of f(f(x))

f(x) =
1) -x-2 when -4<= x <= -1
2) -1 when -1< x <= 1
3) x-2 when 1< x <= 2

How can I start by solving such problem ???
I tried to draw f(x) at first but I don't know what can I do after that

2. ## Re: problem with sketching a graph of composite function

Draw it in "parts" just as it is given. f(x)= -x- 2 is a straight line so determined by two points. Since we only want this for -4<= x<= -1, evaluate at those end points. f(-4)= -(-4)- 2= 4- 2= 2 and f(-1)= -(-1)- 2= -1. Draw a line between the points (-4, 2) and (-1, -1).

For x between -1 and 1, f(x)= -1 is a constant function: long as x is between -1 and 1, y= -1. Draw the horizontal line from (-1, -1) to (1, -1).

For between 1 and 2, f(x)= x- 2. That is again a linear function. f(1)= 1- 2= -1 and f(2)= 2- 2= 0 so draw the line from (1, -1) to (2, 2).

The entire graph is the "broken line" made of those three line segments.

3. ## Re: problem with sketching a graph of composite function

Originally Posted by mido22
graph the function of f(f(x))
Such problems are solved using the method of intent stare™. Draw the original function as described in post #2. Then consider different intervals. For example, when -1 ≤ x ≤ 1, f(x) = 1, and f(1) = -1, so f(f(x)) = -1 on that intervals. When 1 ≤ x, f(x) grows between -1 and 0. But f(y) = -1 when -1 ≤ y ≤ 0, so f(f(x)) = -1 again on that interval. When -4 ≤ x ≤ -3, f(x) decreases with slope -1 between 2 and 1. When 1 ≤ y ≤ 2, f(y) grows with slope 1 between -1 and 0. Viewed otherwise, when y decreases from 2 to 1, f(y) decreases from 0 to -1. Therefore, f(f(x)) decreases from 0 to -1 on [-4, -3]. But after y = f(x) reaches 1, f(y) stop decreasing. And so on.

When you are done, check the graph here.

4. ## Re: problem with sketching a graph of composite function

That's a great explanation thank you very much "emakarov"
thanks also "HallsofIvy" but I think you didn't read my question carefully

5. ## Re: problem with sketching a graph of composite function

Originally Posted by mido22
That's a great explanation thank you very much "emakarov"
thanks also "HallsofIvy" but I think you didn't read my question carefully
Actually I am still reading your question to understand what you realy wanted to do. Is it a composition of function problem?