Domain of function, Co-domain, Range of function - Practice Questions & MCQ

Domain

All possible values of $x$ for $f(x)$ is defined $(f(x)$ is a real number) is known as a domain.
If a function is defined from $A$ to $B$ i.e. $f: A \rightarrow B$, then all the elements of set $A$ is called the Domain of the function.

Co-domain

If a function is defined from $A$ to $B$ i.e. $f: A \rightarrow B$, then set $B$ is called the Co-domain of the function.

Range

The set of all possible values of $f(x)$ for every $x$ belonging to the domain is known as the Range of this function.

For example, let $A=\{1,2,3,4,5\}$ and $B=\{1,4,8,16,27,64,125\}$. The function $f: A->B$ is defined by $f(x)=x^3$. So here,

Domain: Set A

Co-Domain: Set B
Range: $\{1,8,27,64,125\}$

The range is always a subset of the co-domain and the Range can be equal to the co-domain in some cases.

Note: If only the formula is given, then the co-domain is R, and the domain and range have to be found.

Domain in this case will be all the real values of x for which y is real

Range is all the real values of y corresponding to values of x in the domain

Rules to Find Domain
If the domain of $f(x)$ is $A$ and the domain of $g(x)$ is B, then the domain of $f(x)+g(x), f(x)$ $-g(x), f(x) . g(x)$ is $A \cap B$.
For the domain of $f(x) / g(x)$, remove values of $x$ for which $g(x)=0$, from $A \cap B$.
Domain of expressions of type $\sqrt{f(x)}$, we take the common values between A and values of $x$ for which $f(x) \geq 0$.
Domain of the polynomial function is R .
Graphical method: we can also find the domain if only the graph of a function is given. We will learn this through the help of solved examples.

Methods to find Range
Simple manipulations
For the range of $y=f(x)$, we can first express $x$ as a function of $y: x=g(y)$. Now the domain of $x=g(y)$ is the same as the range of $y=f(x)$
Graphical method: we can also find the range if only the graph of a function is given.

We will learn these through the help of solved examples later