Math Is Fun Forum

ziabing

Member

Registered: 2021-09-18

Posts: 6

Composite function : true or false?

Hello!

I'm struggling with the following question :

True or false?
If f,g : R → R are defined as f(x) = |x−1| et g(x) = |x+1| then the composite function g ◦ f verifies, for x ∈ R :
(g ◦ f)(x) = 2-x  if x ≤ 1,
(g ◦ f)(x) = x  if x > 1,

I'm pretty sure that (g ◦ f)= |x-1|+1. However, I don't know how to continue. I've graphed all three functions and the statement seems true to me, but I'm not very confident.

Can someone confirm if the statement is true or false? If it's true, how can I prove it? If it's false, how can I disprove it?

Thanks in advance!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Composite function : true or false?

hi ziabing

Welcome to the forum.

Here's how I tackled this.

I drew a pair of function boxes,; the first showing f; the second g.

Then I tried values of x, starting with x = 1, then 2 then 3 etc.

1 .........f .........0 ......g ......1
2 .........f .........1 ......g ......2
3 .........f .........2 ......g ......3

This is certainly gf (x) = x for x ≥  1

So far so good.

0 ..........f .........1 ......g ......2
-1 .........f .........2 ......g ......3
-2 .........f .........3 ......g ......4
-3 .........f .........4 ......g ......5

That looks good too.

How do you 'prove it'.

For x ≥ 1, f(x) = x-1

So gf(x) = g(x-1) = x-1 + 1 = |x| = x as x is positive in this range.

For x < 1, f(x) = 1 - x

So gf(x) = g(1-x) = |1-x + 1| = |2-x|  As x < 1, 2-x is always positive so the absolute lines are unnecessary, hence gf(x) = 2-x

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ziabing

Member

Registered: 2021-09-18

Posts: 6

Re: Composite function : true or false?

Understood. Thank you so much!