Friday, April 19, 2002

Sketch f(x) = 2|3x-1|-3|x+1|-3?

can you tell me how to sketch this function, step by step? It's a modulus function

Answer on Sketch f(x) = 2|3x-1|-3|x+1|-3?

1)

Consider where the modulus will cause a change of sign.
|3x - 1| causes a change of sign at x = 1/3
|x + 1| causes a change of sign at x = -1

2)

Consider the two points where the sign changes because of the modulus, and build intervals base on them.
The first interval will be x < -1, the second will be -1 <= x <= 1/3, the third will be x > 1/3

3)

Check and see if 3x - 1 AND x + 1 will be positive in this three intervals.

When x < -1, 3x - 1 is negative, x + 1 is negative.
Thus, we let |3x - 1| = -(3x - 1) = 1 - 3x and |x + 1| = -(x + 1) = -x - 1
f(x) = 2(1 - 3x) - 3(-x - 1) - 3 = 2 - 6x + 3x + 3 - 3 = -3x + 2
Hence, for x < -1, we sketch f(x) = -3x + 2

When -1 <= x <= 1/3, 3x - 1 is negative and x + 1 is positive.
Thus, we let |3x - 1| = -(3x - 1) = 1 - 3x and |x + 1| = x + 1
f(x) = 2(1 - 3x) - 3(x + 1) - 3 = 2 - 6x - 3x - 3 - 3 = -9x - 4
Hence, for -1 <= x <= 1/3, we sketch f(x) = -9x - 4

When x > 1/3, 3x - 1 is positive and x + 1 is positive
Thus, we let |3x - 1| = 3x - 1 and |x + 1| = x + 1
f(x) = 2(3x - 1) - 3(x + 1) - 3 = 6x - 2 - 3x - 3 - 3 = 3x - 8
Hence, for x > 1/3, we sketch f(x) = 3x - 8

This is how the graph should look like:
http://www.wolframalpha.com/input/?i=y+%…