# Compose Functions: True/False?

• MHB
• lemonthree
In summary: The latter is required for the composition $h_1\circ h_2$ to be defined but not for $h_1$ and $h_2$ to be equal.The option "If A=B=C, then f∘g=g∘f" is false because it is possible for $f\circ g$ to be defined while $g\circ f$ is not, even if $A=B=C$. This is because the domains of $f$ and $g$ can be different even if their codomains are equal. Therefore, we cannot assume that $f\circ g$ and $g\circ f$ are necessarily equal under the given assumptions. In summary, the first 3 options are
lemonthree
Which of the following are true? Select all options. Assume that f:A→B and g:B→C.

 If f and g are injective, then so is g∘f If f and g are surjective, then so is g∘f If f and g are bijective, then so is g∘f If g∘f is bijective, then so are both f and g If g∘f is bijective, then so is either f or g If A=B=C, then f∘g=g∘f.

The first 3 options are true while the next 2 options are false. The only question I have here is regarding the last option "If A=B=C, then f∘g=g∘f."

Why can't it be true? If A = B = C, and they are all equal sets, we know that f:A→B and g:B→C.
So in essence, isn't f∘g=g∘f?
f∘g= f(g(x)) = f(C) = f(A) = B
While g∘f = g(f(x)) = g(B) = C
But A = B = C so they are all equal.
Range = codomain = domain anyway.

lemonthree said:
The only question I have here is regarding the last option "If A=B=C, then f∘g=g∘f."
Then it makes sense to post only that question.
lemonthree said:
Why can't it be true?
The question is not whether $f\circ g=g\circ f$ can be true, it's whether it must be true under the given assumptions.
lemonthree said:
If A = B = C, and they are all equal sets, we know that f:A→B and g:B→C.
More than that, we are given that $f:A\to B$ and $g:B\to C$ even if $A=B=C$ does not hold.

(A relevant joke. A tourist asks a local: "If I go down this street, will there be a railway station? The local replies, "The station will be there even if you don't go down that street".)

lemonthree said:
f∘g= f(g(x)) = f(C)
$f(g(x))$ cannot equal $f(C)$ because the former is an element of the codomain of $f$ while $f(C)$, if defined, is a subset of the codomain of $f$. In any case, equality of two functions $h_1$ and $h_2$ means that $h_1(x)=h_2(x)$ for every $x$ in the domain of $h_1$ and $h_2$ and not just that the domain and codomain of $h_1$ and $h_2$ are the same.

## 1. What is a composed function?

A composed function is a mathematical concept where one function is applied to the output of another function. It is denoted by (f ∘ g)(x) and read as "f composed with g of x".

## 2. How do you determine if two functions can be composed?

To determine if two functions can be composed, you need to check if the output of the first function is a valid input for the second function. This means that the domains of both functions must align.

## 3. Is the composition of two functions always commutative?

No, the composition of two functions is not always commutative. This means that (f ∘ g)(x) may not always be equal to (g ∘ f)(x). The order in which the functions are composed matters.

## 4. Can you compose more than two functions?

Yes, you can compose more than two functions. For example, if you have three functions f, g, and h, you can compose them as (f ∘ g ∘ h)(x), which means the output of h will be the input for g, and the output of g will be the input for f.

## 5. How do you evaluate a composed function?

To evaluate a composed function, you need to start from the innermost function and work your way out. Substitute the input value into the innermost function, then use the resulting output as the input for the next function, and so on until you reach the outermost function.

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