Domain of a Function

Functions provide different answers on different input values. This is important to note that we need to consider things that are physically possible, unlike imaginary numbers. For example, a ticket costs 50 dollars, you want to buy all your friends the ticket since it is your birthday. Let's say you have 10 friends and that means the total cost will be 500 dollars. What just happened right now? How you figured that it will cost that much? Basically, you made a function which is f(x) = 50x and since the value of x is 10 hence, f(10) = 50(10) = 500. In other words, if you input 10 in the function, it will give you 500. Hence, the value of 10 gives an answer that will be different from other answers. Although, this isn't something to consider inevitable. That is why mathematicians introduce a new concept of domain and range of a function.

Basically, the domain is the set of elements that has an image. Although the image is a very generic word, in the mathematics world we call it range. Let's take an example below:

f(x) = \sqrt { x }

D = {x /f(x)}

Initial set Final set

Domain Range

 

The left oval represents the domains of the function and on right are the ranges of the function. You might be wondering why there is no image of -2, -1? That is because it will result in an imaginary number. Hence, we can conclude that the domain of a function is an independent variable, however, the range is the dependent variable. The range shows what you can expect from a function.

The best Maths tutors available
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
1st lesson free!
Intasar
4.9
4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Paolo
4.9
4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Rajan
4.9
4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!
Farooq
5
5 (13 reviews)
Farooq
£35
/h
1st lesson free!
Myriam
5
5 (15 reviews)
Myriam
£20
/h
First Lesson Free>

Different Types of Domains for Different Function

For every function, there can be different domains as well as ranges. Let's talk about them in detail.

Domain of a Polynomial Function

The domain is R.

f(x) = { x }^{ 2 } - 5x + 6 \qquad D = R

Domain of a Rational Function

The domain is R, minus the values that would annul the denominator (there cannot be a number whose denominator is zero).

f(x) = \frac { 2x - 5 }{ { x }^{ 2 } - 5x + 6 }

{ x }^{ 2 } - 5x + 6 \qquad D = R - \left \{ 2, 3 \right \}

Domain of an Irrational Function of Odd Index

The domain is R.

f(x) = \sqrt[3]{ { x }^{ 2 } - 5x + 6 } \qquad D = R

f(x) = \sqrt[3]{ \frac { x }{ { x }^{ 2 } - 5x + 6 } } \qquad D = R - \left \{ 2, 3 \right \}

Domain of an Irrational Function of Even Index

The domain is formed by all the values that make the radicand greater than or equal to zero.

f(x) = \sqrt { { x }^{ 2 } - 5x + 6 }

{ x }^{ 2 } - 5x + 6 \geq 0 solve the inequality.

D = \left ( - \infty, 2 \right ] U \left [ 3, \infty \right )

f(x) = \frac { \sqrt { { x }^{ 2 } - 5x + 6 } }{ x + 4 }

{\begin{matrix} { x }^{ 2 } - 5x + 6 \geq 0  \qquad \left ( - \infty, 2 \right ] U \left [ 3, \infty \right ) \\ x + 4 \neq 0 \qquad x \neq -4 \end{matrix}\right

D = \left ( - \infty, -4 \right ] U \left ( -4, 2 \right ] U \left [ 3, \infty \right )

f(x) = \frac { x + 4 }{ \sqrt { { x }^{ 2 } - 5x + 6 } }

{ x }^{ 2 } - 5x + 6 > 0 \qquad D = \left ( - \infty, 2 \right ] U \left [ 3, \infty \right )

f(x) = \sqrt { \frac { x + 4 }{ { x }^{ 2 } - 5x + 6 } }

\frac { x + 4 }{ { x }^{ 2 } - 5x + 6 } \geq 0 solve the inequality.

Domain of a Logarithmic Function

The domain is formed by all the values that make the inside of the logarithmic greater than zero.

f(x) = \log ( { x }^{ 2 } - 5x + 6 )

{ x }^{ 2 } - 5x + 6 > 0 \qquad D = \left ( - \infty, 2 \right ] U \left [ 3, \infty \right )

Domain of an Exponential Function

The domain is R.

Domain of the Sine function

The domain is R.

Domain of a Cosine function

The domain is R.

Domain of a Tangent Function

D = R - \left \{ (2k + 1) . \frac { \pi }{ 2 }; k \in Z \right \}

D = R - \left \{ ..., - \frac { \pi }{ 2 }, \frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }, ... \right \}

Domain of a Cotangent Function

D = R - \left \{ ..., - \pi, 0, \pi, ... \right \}

Domain of a Secant Function

D = R - \left \{ (2k + 1) . \frac { \pi }{ 2 }; k \in Z \right \}

D = R - \left \{ ..., - \frac { \pi }{ 2 }, \frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }, ... \right \}

Domain of a Cosecant Function

D = R - \left \{ k \pi ; k \in Z \right \}

D = R - \left \{ ..., - \pi, 0, \pi, ... \right \}

Domain of Operations with Functions

D(f + g) = D(f - g) = D(f . g) = D(f) \cap D(g)

D(\frac { f }{ g }) = D(f) \cap D(g) - \left \{ x \in \mathbb{R} / g(x) =  0 \right \}

f(x) = \frac { \sqrt { x + 4 } }{ \sqrt { { x }^{ 2 } - 5x + 6 } }

\left\{\begin{matrix} x + 4 \geq 0 \qquad \left [ -4, \infty ) \\ { x }^{ 2 } - 5x + 6 > 0 \qquad ( - \infty, 2) U ( 3, \infty) \end{matrix}\right

Solved the system of Inequalities.

D = \left [ -4, 2 \right ) U (3, \infty)

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 3 vote(s)
Loading...

Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.