Inverse function of an absolute

Loki123 said:
How do I find the inverse of an absolute?
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You don't ;) But your function can be inverted nevertheless: just write an equation ([imath]|x+1|+|2x+4| + 4x = a[/imath]) and solve it for x, which will give you [imath]x[/imath] as a function of [imath]a[/imath].
 
blamocur said:
You don't ;) But your function can be inverted nevertheless: just write an equation ([imath]|x+1|+|2x+4| + 4x = a[/imath]) and solve it for x, which will give you [imath]x[/imath] as a function of [imath]a[/imath].
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but won't i have multiple possibilities?
 
Loki123 said:
but won't i have multiple possibilities?
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I believe you will not. If you draw the function's graph you will see that it is piecewise linear and monotone. Solving it separately for each of the linear pieces will get you the answer.
 
blamocur said:
I believe you will not. If you draw the function's graph you will see that it is piecewise linear and monotone. Solving it separately for each of the linear pieces will get you the answer.
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I got three answers for x. What now?
 
[imath]f(x) = |x+1| + 2|x+2| + 4x[/imath]

[imath]x \ge -1 \implies f(x) = 7x+5 \implies f(-1)=-2 \implies f^{-1}(-2) = -1 \text{ and } f(0)=5 \implies f^{-1}(5) = 0[/imath]

[imath]x \le -2 \implies f(x) = x-5 \implies f(-2) = -7 \implies f^{-1}(-7) = -2[/imath]
 
skeeter said:
[imath]f(x) = |x+1| + 2|x+2| + 4x[/imath]

[imath]x \ge -1 \implies f(x) = 7x+5 \implies f(-1)=-2 \implies f^{-1}(-2) = -1 \text{ and } f(0)=5 \implies f^{-1}(5) = 0[/imath]

[imath]x \le -2 \implies f(x) = x-5 \implies f(-2) = -7 \implies f^{-1}(-7) = -2[/imath]
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hm interesting, give me a minute to see if i get this.
 
skeeter said:
[imath]f(x) = |x+1| + 2|x+2| + 4x[/imath]

[imath]x \ge -1 \implies f(x) = 7x+5 \implies f(-1)=-2 \implies f^{-1}(-2) = -1 \text{ and } f(0)=5 \implies f^{-1}(5) = 0[/imath]

[imath]x \le -2 \implies f(x) = x-5 \implies f(-2) = -7 \implies f^{-1}(-7) = -2[/imath]
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yeah i don't get it. how did you know in which one to put what values to get what you need for inverse functions???
 
Loki123 said:
yeah i don't get it. how did you know in which one to put what values to get what you need for inverse functions???
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Write it out as a piece-wise function.
[math]f(x) = \begin{cases} x-5 &\text{if } x\le -2 \\ 5x+3 &\text{if } -2 < x<-1\\ 7x+5 &\text{if } x \ge -1\\ \end{cases}[/math]
 
I think that you can do this by inspection only.

Since the first two terms are never negative, the only way for f(x) = -7 is for the last term, 4x, to be negative. 4x<0, whenever x<0.

I would try x=-1 first. This surely will not work since 4x will be -4 which is not negative enough.
Now try x=-2. Wow it works!

To find x such that f(x)=-2:
Since -2>-7, I would try x values that are negative but bigger that -2. Maybe x=-1 will work? Does it.


To find x such that f(x)=5:
Since only 4*0<5 and 4*1<5 I would try non-negative values x=0 and x=1. Do any of these work?
 
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