Mathaversity Tutors

mathaversity
mathaversity
Menu
Get Started

Functions and Their Graphs

Introduction

The idea of a function is very important in math, especially in AP Calculus AB. Limits, continuity, derivatives, and applications are all based on knowing how functions and their graphs work. Students who have a good understanding of different types of functions and how their graphs work can better understand mathematical models and solve real-world problems quickly.

This guide gives a full and detailed explanation of common functions and their graphs, including a clear mathematical definition, how they behave on a graph, their most important properties, and how they relate to AP Calculus AB.

What Is a Function?

A function is a rule that assigns exactly one output to each input.

Mathematically, a function is written as: f :\mathbb{R} \rightarrow \mathbb{R}

\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad y = f(x)

  • Here:
  • x is the independent variable
  • y is the dependent variable
  • The set of all possible inputs is called the domain
  • The set of all possible outputs is called the range
  • A graph represents a function visually by plotting all ordered pairs (x,f(x)).

Identity Function

A function is called an identity function if each element of the domain is mapped to itself.

  • Properties
  • Domain: \mathbb{R}
  • Range: \mathbb{R}
  • One-to-one and onto
identity function

Constant Function

A function is called a constant function if it is of the form: f(x) = c. where c is a real constant.

  • Properties
  • Domain: \mathbb{R}
  • Range: \{c\}
  • No increase or decrease
  • The graph of a constant function y = f(x) = 3 is given below.
constant function

Polynomial Functions

A function f : \mathbb{R} \rightarrow \mathbb{R} is said to be a polynomial function if for each x \in \mathbb{R}, f(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n

where:

  • n is a non-negative integer
  • a_0, a_1, a_2, \dots, a_n \in \mathbb{R}
  • a_n \neq 0
  • Polynomial functions are continuous and smooth for all real values of x.
polynomial function

Rational Functions

A function is called a rational function if: f(x) = \frac{p(x)}{q(x)}. where p(x) and q(x) are polynomial functions and q(x) \neq 0. The example graph of a rational function is given below:
Rational Functions

Absolute Value Function

The absolute value function is defined as: f(x) = |x|

  • Piecewise Form
The absolute value function is defined as: |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}

  • Properties
  • Domain: \mathbb{R}
  • Range: |y \ge 0|
  • Not differentiable at x = 0
  • The graph of absolute value function y = |x| is given below.
absolute value function

Exponential Functions

A function of the form: f(x) = a^x, \quad a > 0, \ a \neq 1 is called an exponential function.

  • Properties
  • Domain: \mathbb{R}
  • Range: y > 0
  • Horizontal asymptote: y = 0
Exponential Functions

Conclusion

mathaversity logo