Table of Contents
What is a Function in Math?
A function in math is a set of ordered pairs where each element in the set corresponds to a unique output. The function assigns a unique output to every input.
Types of Functions in Mathematics with Examples
A mathematical function is a set of ordered pairs (x, y) where each x corresponds to a unique y. In other words, a function is a relationship between two variables in which each value of x corresponds to a unique value of y.
There are many different types of functions in mathematics, but some of the most common ones include linear functions, exponential functions, and logarithmic functions.
Linear Functions
A linear function is a function in which the graph is a straight line. The equation for a linear function can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Some examples of linear functions include y = 2x + 1, y = -3x + 5, and y = x + 1.
Exponential Functions
An exponential function is a function in which the graph is a curve that increases or decreases at an exponential rate. The equation for an exponential function can be written in the form y = a xb, where a is the initial value, b is the growth or decay factor, and x is the time variable.
Some examples of exponential functions include y = 5x, y = 2x – 3, and y = 0.5x.
Logarithmic Functions
A logarithmic function is a function in which the graph is a curve that increases or decreases at a
Types of Function – Based on Elements
There are three types of functions – based on elements:
1. Functions based on numbers
2. Functions based on variables
3. Functions based on sets
Types of Function – Based on Equation
A polynomial function is a function that is defined by a polynomial equation. A polynomial equation is an equation that has a polynomial function on one side and one or more constants on the other side.
A rational function is a function that is defined by a rational equation. A rational equation is an equation that has a rational function on one side and one or more constants on the other side.
A radical function is a function that is defined by a radical equation. A radical equation is an equation that has a radical function on one side and one or more constants on the other side.
Types of Functions – Based on Range
There are four types of functions based on their range:
1. One-to-One Functions
A one-to-one function is a function where every input corresponds to a unique output. There is no “duplicate” output.
For example, the function f(x) = x² is one-to-one because for every input x, there is a unique output f(x) = x². No matter what value x we input, the output will always be different.
2. Onto Functions
An onto function is a function where every output corresponds to a unique input. There is no “duplicate” input.
For example, the function f(x) = x is onto because for every output f(x), there is a unique input x. No matter what value f(x) we input, the output will always be different.
3. Inverse Functions
An inverse function is a function where every input corresponds to a unique output, and vice versa.
For example, the function f(x) = x² is inverse to the function g(x) = x because for every input x, the output of f(x) is the input of g(x). And for every input x, the output of g(x) is the input of f(x).
4. Bijective Functions
A bijective function
Types of Functions – Based on Domain
and Range
The following are six types of functions, based on their domain and range:
1. Domain: real numbers; Range: real numbers
This is the most basic type of function. It can be represented by a graph that consists of a single line.
2. Domain: real numbers; Range: positive real numbers
This type of function is similar to the basic type, except that the range now consists of only positive real numbers.
3. Domain: real numbers; Range: negative real numbers
This type of function is similar to the basic type, except that the range now consists of only negative real numbers.
4. Domain: real numbers; Range: all real numbers
This type of function is similar to the basic type, except that the range now consists of all real numbers.
5. Domain: all real numbers; Range: all real numbers
This type of function is the most general type. It can be represented by a graph that consists of a single curved line.
6. Domain: all real numbers; Range: one or more real numbers
This type of function is similar to the basic type, except that the range now consists of one or more real numbers.
Solved Example of Functions
A function is a set of ordered pairs (x, y) where each x corresponds to a unique y.
The following is an example of a function:
x y
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
Representation of Functions
Representing a function
There are a few ways to represent a function. One way is to use a function notation, which is an equation that uses the function name followed by the input and output values separated by a colon. Another way is to use a graph, which is a visual representation of how the function behaves.
Practice Problems
1. Find the derivative of
f(x) = x3
3×2
2. Find the derivative of
f(x) = 2x
2x
3. Find the derivative of
f(x) = x
1
4. Find the derivative of
f(x) = (x+1)3
3×2 + 3x
5. Find the derivative of
f(x) = (x-1)3
3×2 – 3x
