# Greatest Integer Functions and odd, even functions

• Hydroxide
In summary, the homework statement is "If f is even and g is odd, is fog even, odd or neither." The Attempt at a Solution states "Let f(-x)=f(x) and g(-x)=-g(x)". Both of these statements lead to the following equation: fog(-x)=f(g(-x))=f(-g(x))=f(gx). Therefore, fog is even.

## Homework Statement

Q1
[|x+2|]=2[|x|]-3

Q2
If f is even and g is odd, is fog even, odd or neither

## The Attempt at a Solution

Q1
Not sure. Can someone please give me a start on this?
I think if I knew some properties of greatest integer functions I could work it out

Q2
Let f(-x)=f(x) and g(-x)=-g(x)

All I need to know is what fog(-x) equals. The rest I can do myself.

Thanks a lot.

Q1, case work first to take care of the absolute value (x>=0 and x<0).
notice that if n is in integer,
[x+n]=[x]+n

Q2. fog(-x)=f(g(-x)) by definition.

edit: sry typo, it should be x>=0 and x<0

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tim_lou said:
Q1, case work first to take care of the absolute value (n>=0 and n<0).
notice that if n is in integer,
[x+n]=[x]+n

Q2. fog(-x)=f(g(-x)) by definition.

But that leaves me with [|x+2|]=[|2x-3|] correct?
So can i say [|x-5|]=0 ? hmmm, seems to simple

Or you could draw the graphs and find the solution graphically if you know some curve sketching.

fog(-x)=f(g(-x))=f(-g(x))=f(gx)
Hence fog is even.

chaoseverlasting said:
fog(-x)=f(g(-x))=f(-g(x))=f(gx)
Hence fog is even.

Yeah I figured that one out, but thanks anyway

Hydroxide said:
But that leaves me with [|x+2|]=[|2x-3|] correct?
So can i say [|x-5|]=0 ? hmmm, seems to simple

No!
$$2[x]\neq [2x]$$
furthermore,
$$|x|-3 \neq |x-3|$$
watch what you are dealing with, greatest integer function and absolute values have different properties.
$$[x]+[y]\neq [x+y]$$
watch what identities you are using! I suggest you get a deep understand of what greatest integer function means. [x] is the greatest integer that is less than or equal to x.
for instance, [1.01]=1 and [1.9999999999999999999]=1.

suppose you have [1.5]=1, but 2[1.5]=2 while [2*1.5]=3.
for absolute value, suppose you have -3, |-3|=3, but |-3|+2=5, while |-3+2|=1. they are not equal!

get difference cases, for x>0, |x|=x, that is one case
for x<=0, |x|=-x, that is another case, similarly for |x+2|.

get rid of the absolute values first, break it up into two equations.

then get rid of complications in [], since it is not easy to work with [x]. so make [x+2]=[x]+2. the rest is simple algebra.

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## 1. What is a Greatest Integer Function?

A Greatest Integer Function, also known as the Floor Function, rounds down a given number to the nearest integer. For example, the greatest integer function of 3.8 would be 3, and the greatest integer function of -2.5 would be -3.

## 2. How is a Greatest Integer Function represented mathematically?

The Greatest Integer Function is typically represented using the symbol ⌊x⌋, where x is the input number. It can also be written as floor(x) or [x].

## 3. What is an Odd Function?

An Odd Function is a function where f(-x) = -f(x), meaning that the function's output is the negative of its input when the input is negated. Visually, an odd function is symmetric about the origin on a coordinate plane.

## 4. What is an Even Function?

An Even Function is a function where f(-x) = f(x), meaning that the function's output remains the same when the input is negated. Visually, an even function is symmetric about the y-axis on a coordinate plane.

## 5. How can I identify if a function is odd, even, or neither?

To determine if a function is odd, even, or neither, you can use the properties of odd and even functions. If f(-x) = -f(x), the function is odd. If f(-x) = f(x), the function is even. If neither of these properties hold, the function is neither odd nor even.