Question on functions (absolute fns.)

[rock.freak667]

Homework Statement

$$f:\rightarrow x+1,g:x\rightarrow |x|$$

Solve the equation gf(x)=fg(x)

Homework Equations

The Attempt at a Solution

gf(x)=|x+1|
and fg(x)=|x|+1

so I drew the graphs of y=gf(x) and y=fg(x) on the same axes.
For x<0 the graphs do not intersect as the two lines are parallel (having the same gradient) and hence there is no solution for x<0.

BUT, for x>0, the two lines are the same...so that means there are an infinite number of solutions for x>0. Does that mean I write the answer as {x:x>0} ?

 

[HallsofIvy]

Almost. If x= 0, |0+1|= 1 so x= 0 also satisfies the equation. In particular, if x> 0, |x|+ 1= x+ 1 and since x>0>-1, |x+1|= x+ 1 so the equation |x|+1= |x+1| is the same as x+1= x+1. That's satisfied for all x so |x+ 1|= |x|+ 1 is satisfied for all $x\ge 0$ (not "x> 1").

 

[Architeuthis Dux]

Yes, you are correct. The solution to this equation is {x:x>0}. This means that any value of x greater than 0 will satisfy the equation. This is because for x>0, both functions will have the same value, making them equal to each other. However, for x<0, the two functions will never intersect, so there is no solution for x<0. This is a good visualization of the concept of absolute functions and how they behave differently for positive and negative values. Well done!