Graphs of Functions: The proverb, “I hear I forget, I see I remember, I do I understand”, rightly emphasizes the importance of viewing the concepts for a better understanding. Even abstract concepts like functions can get interesting when they are made using images. In such a scenario, the graphical representations of functions give an interesting visual treat and a strong theoretical ground. It not only helps us identify if a graph is a function, but it also clarifies characteristics of functions such as monotonicity and odd-even nature.

The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation. This article will take you through various types of graphs of functions.

Definition of Graph of a Function

A relation \(f\) from set \(A\) to set \(B\) is a function if every element of set \(A\) has only one image in set \(B.\) It is a subset of \(A \times B.\) The set of elements in \(A\) that are plugged into the function \(f\) is called the domain. The set of images of the elements in \(A,\) a subset of \(B\) is called the range of the function \(f.\)
Here, the relation \(R\) is a function from the set \(X\) to\(Y\):

Graph of a Function

The graph of a function \(f\) is the graph of the equation \(y = f\left( x \right).\) In other words, it is the set of all points \(\left( {x,\,f\left( x \right)} \right).\)

Some Functions and Their Graphs

1. Constant Function

A function \(f:R \to R\) defined by \(f\left( x \right) = c,\,\forall x \in R\) is called a constant function. For example, the function \(f:R \to R\) defined by \(f\left( x \right) = 3,\,\forall x \in R\) is graphed as:

2. Identity Function

A function \(f:R \to R\) defined by \(f\left( x \right) = x,\,\forall x \in R\) is called an identity function. An identity function is graphed as:

3. Linear Function

A function \(f:R \to R\) of the form \(f\left( x \right) = ax + b,\,\forall x \in R\) is called a linear function. For example, the function \(f:R \to R\) defined by \(f\left( x \right) = 2x + 3,\,\forall x \in R\) is graphed as: Note that an identity function is a linear function with \(a = 1\) and \(b = 0.\)

4. Modulus Function

A function \(f:R \to R\) defined by \(f\left( x \right) = \left| x \right|,\,\forall x \in R\) is called the modulus function. For negative values of \(x,\,f\left( x \right) = \, – x\) and for non-negative values \(f\left( x \right) = x.\)
The graph of the function \(f\left( x \right) = \left| x \right|\) is drawn as:

This function is also called the absolute value function.

5. Greatest Integer Function

A function \(f:R \to R\) defined by \(f\left( x \right) = \,\left[ x \right],\,\forall x \in R\) where \(\left[ x \right]\) represents the greatest integer less than the real number \(x,\) and this is called the greatest integer function. The symbol \(\left\lfloor x \right\rfloor \) is also used for the same.
The domain of the greatest integer function is the set of real numbers, and the range is a set of integers.
The graph of the greatest integer function is drawn as:

This function is also called the floor function.

6. Smallest Integer Function

A function \(f:R \to R\) defined by \(f\left( x \right) = \left\lceil x \right\rceil ,\,\forall x \in R,\) where \(\left\lceil x \right\rceil \) represents the smallest integer greater than the real number \(x\) is called the smallest integer function.
As in the case of the greatest integer function, the domain of the smallest integer function is the set of real numbers, and the range is a set of integers.
The graph of the smallest integer function is drawn as:

This function is also called the ceiling function.

7. Signum Function

A function \(f:R \to R\) defined by
\(f\left( x \right) = \left\{ {\frac{{\left| x \right|}}{x},\,\forall x \ne 0\,0,\,\forall x = 0} \right.\)
is called the signum function.
Another definition for the same is given by:
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} 1&{x > 0}\\ \begin{array}{l} 0\\ – 1 \end{array}&\begin{array}{l} x = 0\\ x < 0 \end{array} \end{array}} \right.\)
The graph of the smallest integer function is drawn as:

The domain of the signum function is the set of real numbers, while the range is a finite set \(\left\{ { – 1,\,0,\,1} \right\}.\)

8. Exponential Function

A function \(f:R \to R\) defined by \(f\left( x \right) = {a^x},\,\forall x \in R,\) and \(a > 0,\,a \ne 1\) is called an exponential function. For the values of \(a > 1,\) the function’s value increases exponentially, and for \(0 < a < 1,\) it approaches zero but never touches it.
The graph of the function \(f\left( x \right) = {a^x}\) is drawn as:

9. Logarithmic Function

For \(a > 0\) and \(a \ne 1,\) a function \(f:\left( {0,\,\infty } \right) \to R\) defined as \(f\left( x \right) = x,\,\forall x > 0,\) is called logarithmic function. For the values of \(a > 1,\) the function’s value increases exponentially, and for \(0 < a < 1,\) it decreases.
The graph of the function \(f\left( x \right) = x\) is drawn as:

The logarithmic and exponential functions are inverses of each other.

10. Polynomial Function

A function \(f:R \to R\) is said to be a polynomial function if it is defined as \(f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + … + {a_n}{x^n},\,\forall x \in R,\,n\) is a non-negative integer and \({a_0},\,{a_1},\,{a_2},\,…{a_n} \in R.\)
For example, the graph of the polynomial function \(f\left( x \right) = {x^3} + 2{x^2} – 1\) is drawn as:

11. Rational Function

Rational functions have a form \(\frac{{f\left( x \right)}}{{g\left( x \right)}},\) where \({f\left( x \right)}\) and \({g\left( x \right)}\) are polynomial functions and \(g\left( x \right) \ne 0.\)
For example, the graph of the rational function \(f\left( x \right) = \frac{{x + 1}}{{{x^2} – 2}}\) is drawn as:

We have seen the graphs of different types of functions. Now, if we have a graph, is it possible to check the type of that function?

Vertical Line Test

The vertical line test is a visual method to determine whether the given graph is of a function or not. A graph represents a function if no vertical line intersects the graph at more than one point. For example, the first graph represents a function, whereas the second one does not!

Characteristics of Graphs

1. Increasing and Decreasing Functions

As the name suggests, a function is said to be increasing when the value of the dependent variable \(y\) increases with \(x.\)
Consider the graph of the function \(f\left( x \right) = 3x.\)

It is obvious looking at the graph that, as the value of \(x\) increases, the corresponding \(y\) values also increase. So, \(y\) is an increasing function.
A linear function with a positive slope is always an increasing function.
Contrary to the increasing functions, a function is said to be decreasing when the value of the dependent variable \(y\) decreases as \(x\) increases.
Consider the function. Observe that, as the value of \(x\) increases from \( – 2\) to \( 2,\) the corresponding \(y\)- value decreases from \(6\) to \(-6.\) So, \(g\left( x \right)\) is a decreasing function.
A linear function with a negative slope is always an increasing function.
Some functions may be increasing or decreasing at particular intervals.

2. Odd and Even Functions

A function is said to be even if \(f\left( { – x} \right) = f\left( x \right)\) for all \(x\) in the domain. Geometrically, this can be explained using symmetry. That is, a function is an even function if its graph is symmetric about the \(y\)-axis.
A function is said to be even if \(f\left( { – x} \right) = \, – f\left( x \right)\) for all \(x\) in the domain. That is, a function is an even function if its graph is symmetric about the origin.

Solved Examples

Q.1. Which of the following is the graph of the function \(f\left( x \right) = \left| {x – 1} \right|\)?

Ans: The graph of a modulus function is a V-shaped straight curve. The parent modulus function \(f\left( x \right) = \left| x \right|\) has the tip at the origin, and the function \(f\left( {x – 1} \right)\) is obtained by translating the entire curve one unit to the right. Thus, the required graph is the one in option (iii).

Q.2. What is the domain of the function \(f\left( x \right) = \sqrt {64 – {x^2}} \)?
Ans: For the function to be defined, the expression inside the square root symbol needs to be greater than or equal to zero.
That is, \(64 – {x^2} \ge 0\)
\(64 \ge {x^2}\) or \({x^2} \le 64\)
\( – 8 \le x \le 8\)
Therefore the domain of the function is \(\left\{ {x \in R,\, – 8 \le x \le 8} \right\}\) or the closed interval \(\left[ { – 8,\,8} \right].\)

Q.3. Does this graph represent a function? Explain.

Ans: The vertical line test can be used to identify whether a graph is a function. The test states that a graph is of a function if no vertical line intersects the graph at more than one point.

Since any vertical line would intersect the given graph only at one point each, the graph is of a function.

Q.4. Draw the graph of the function defined by \(f\left( x \right) = {x^2} – 4.\)
Ans: Find the values of the function for various values of \(x.\)

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f\left( x \right)\)	\(5\)	\(0\)	\(-3\)	\(-4\)	\(-3\)	\(0\)	\(5\)

Plot the points and join them by a smooth curve.

Q.5. Look at the graph of the function and identify whether this function is even, odd or neither.

Ans: The graph of the function is symmetric about the origin. Therefore, the function is odd.

Q.6. Sketch the graph of the function \(f\left( x \right) = {\left( {x – 1} \right)^2} + 1\) and show that \(f\left( p \right) = f\left( {2 – p} \right).\) Illustrate this result on your graph by choosing one value of \(p.\)
Ans: Find the values of the function for various values of \(x.\)

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(f\left( x \right)\)	\(10\)	\(5\)	\(2\)	\(1\)	\(2\)	\(5\)	\(10\)

Plot the points and join them by a smooth curve.

Now,
\(f\left( {2 – p} \right) = {\left( {2 – p – 1} \right)^2} + 1\)
\( = {\left( {1 – p} \right)^2} + 1\)
\( = {\left( { – 1} \right)^2}{\left( {p – 1} \right)^2} + 1\)
\( = {\left( {p – 1} \right)^2} + 1\)
\( = f\left( p \right)\)
For example, consider the plotted point for the value \(p = 0.\)
Here, \(f\left( p \right) = f\left( 0 \right) = f\left( {2 – p} \right) = f\left( 2 \right) = 2.\)

Summary

The article helps you understand the basics of functions and the ways to represent those using graphs. The article also discusses the definition and the graphs of various functions such as constant, linear, identity, modulus, greatest integer function, smallest integer function, signum function, exponential, logarithmic, rational and polynomial functions.

Further, it explores the vertical line test, which helps you classify a relation into a function. Later the article explains a few characteristics of a function visible in their graphs and concludes with a few solved examples to strengthen the concepts discussed.

Frequently Asked Questions

Below are some Frequently Asked Questions from Graphs of Functions.

Q.1. What is a function rule?
Ans: When a function is defined from a set \(A\) to set \(B,\) the image of an independent variable is related to the independent variable with some set rules. This rule is called the function rule. For example, for the function \(f\left( x \right) = 3x – 5\) the rule is to multiply the variable by \(3\) and subtract \(5.\)

Q.2. How do you find the function of a graph?
Ans: Functions are defined with a function rule – the rule with which the dependent and the independent variables are related. The graph of a function \(f\) is the graph of the equation \(y = f\left( x \right).\) That is, it is the set of all points \(\left( {x,\,f\left( x \right)} \right).\) So, the function rule can be identified from the points on a graph as each point has the values of dependent and independent variables that are related to each other via that function rule, thus identifying the function.

Q.3. How do you know if a graph is a function?
Ans: To understand whether a graph is a function, the vertical line test is used. The test states that a graph is of a function if no vertical line intersects the graph in more than one point.

Q.4. Is a circle on a graph a function?
Ans: The vertical line test can be used to find whether a graph is a function or not. A graph is that of a function if and only if no vertical line intersects the graph at more than one point. For a circle, you can draw a vertical line that intersects at two points, and hence it is not a function.

Q.5. Can a straight line be a function?
Ans: Yes, a straight line is a linear or constant function with any vertical line intersecting exactly at one point.