Topic 1.3 Mathematical Functions

Question 1

What is domain, codomain and range?

Answer 1

What can go into a function is called the domain (a nonempty set A)
What may potentially come out of a function is called the codomain (a nonempty set B)
What actually comes out of a function is called the range or image (the set of elements that get pointed to in B)

Question 2

Is a function the same as a rule?

Answer 2

No. A function needs three parts: an input (domain), and output (codomain) and a subset of A x B such that each element of A appears as the first element in exactly one ordered pair (i.e. the rule; subset of input-outputs). Without these three things, you cannot refer to something as a function

Question 3

Does every output need to be defined in a function?

Answer 3

Not every output needs to be defined for it to be a function, but if the statement doesn’t specify an output for a value in the domain, then it is no longer a function.

Question 4

In functions, can you have two of the same inputs? Can you have two of the same outputs?

Answer 4

Each input can only have one defined output, otherwise the function is not well defined and not a function at all.
However, you can have two of the same input, so long as you get two different outputs at the end

Question 5

If two functions have the same rule, that means they are the same function.

Answer 5

False, if you change any component of the function, be it the rule, the domain or the codomain, then you have a completely different function.
E.g. The follow functions are different
f:R→R, f(x)=sin⁡(x)
g:R→[−1,1], g(x)=sin⁡(x)

Question 6

Provide the general formula to find the range of f: A–> B, where A and B are sets, in both set builder notation and abbreviated set builder notation

Answer 6

SBN: range (f)={b∈B|there exists a∈B such that f(a)=b}
Abbrev. SBN: range (f)={f(a)|a∈A}

Question 7

What is the use of codomains?

Answer 7

We are pretty frequently dealing with functions that take real numbers as outputs, so it is generally easier to just use real numbers as the codomain instead of dedicating time to try and solve the range. Additionally:
- May be hard to compute the range (f)
- Potentially may want to compare multiple functions f, g, so treat with the same domain and codomain
- Might want your codomain to represent more information about the function

Question 8

Find the range of sin(x) when we restrict the domain to the set C={kπ|k∈Z)

Answer 8

Basically, be asked to find the output for each element of the domain, which has now been restricted.
Since f(kπ)=0 for all k∈Z, range(f)={0}

Question 9

Let A and B be sets, and let f: A –> B. Let S be a subset of A. Express the image of S under f in both set builder notation and abbreviated set builder notation

Answer 9

f(S)={b∈B│there exists s∈S such that f(s)=b}
f(S)={f(s)|s∈S}

Question 10

If you are given a graph with a function f: R→R and you are then asked to find some f([a,b]) what is it that you should do?

Answer 10

You are being asked to find the codomain, so should look at the image and see which values would constitute that.
E.g. You are given a sin function between 0 and π, and are then asked to find f([0,π]).
Your answer would look like f([0,π]) = [0,1]

Question 11

Let A be a subset of R. Let f: A –> R and g: A –> R. How do you define the function f+g with codomain equal to R? What is the domain?

Answer 11

f+g:A→R f+g=f(x)=g(x). The domain is the set of A

Question 12

Let A be a subset of R. Let f: A –> R and g: A –> R. How do you define the function f-g with codomain equal to R? What is the domain?

Answer 12

f−g:A→ R (f−g)(x) = f(x)−g(x). The domain is the set of A

Question 13

Let A be a subset of R. Let f: A –> R and g: A –> R. How do you define the function fg with codomain equal to R? What is the domain?

Answer 13

fg:A→R (fg)(x) =f(x)∙g(x). The domain is the set of A

Question 14

Let A be a subset of R. Let f: A –> R and g: A –> R. How do you define the function f/g with codomain equal to R? What is the domain?

Answer 14

f/g:A→ R (f/g)(x)=f(x)/g(x) The domain is f/g:{a∈A│g(a)≠0}→R

Question 15

Let A and B be a subset of R. Let f: A –> R and g: B –> R. Can you do f+g? Can the problem be resolved at all?

Answer 15

You cannot add together f and g, since they have different domains. However, can resolve it by going f+g:A∩B→R

Question 16

What does the ‘∘’ in ‘f∘g’ mean?

Answer 16

It stands for composite, the symbol is read as ‘circle’, and the statement is read as ‘composite of ‘f’ with ‘g’’. It means the output of function ‘g’ is the input for function ‘f’

Question 17

What is a good way to think of composition?

Answer 17

To think of it as substitution

Question 18

Let f, g: R→R be respectively given by f(x)=2x+1 and g(x)=x^2+3. Find formula for f(g(x)).

Answer 18

(f∘g)(x)=f(g(x))
=f(x^2+3)
=2(x^2+3)+1

Question 19

Let f, g: R→R be respectively given by f(x)=2x+1 and g(x)=x^2+3. Find formula for g(f(x)).

Answer 19

(g∘f)(x)=g(f(x))
=g(2x+1)
=〖(2x+1)〗^2 + 3