Identifying Graphs of Linear, Quadratic, Cubic, and Reciprocal Functions

10th Mathematics : UNIT 1 : Relation and Function

Identifying the graphs of Linear, Quadratic, Cubic and Reciprocal functions

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Graphs provide visualization of curves and functions. Hence, graphs help a lot in understanding the concepts in a much efficient way.

In this section, we will be discussing about the identification of some of the functions through their graphs. In particular, we discuss graphs of Linear, Quadratic, Cubic and Reciprocal functions.

1. Linear Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = mx + c$, $m \neq 0$ is called a linear function. Geometrically this represents a straight line in the graph.

Some Specific Linear Functions and their graphs are given below.

2. Modulus or Absolute valued Function

A function $f: \mathbb{R} \rightarrow [0, \infty)$ defined by $f(x) = |x|$.

Note

• Modulus function is not a linear function but it is composed of two linear functions x and –x.
• Linear functions are always one-one functions and has applications in Cryptography as well as in several branches of Science and Technology.

3. Quadratic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^2 + bx + c$, $(a \neq 0)$ is called a quadratic function.

Some specific quadratic functions and their graphs:

The equations of motion of a particle travelling under the influence of gravity is a quadratic function of time. These functions are not one – one. (Why?)

4. Cubic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^3 + bx^2 + cx + d$, $(a \neq 0)$ is called a cubic function. The graph of $f(x) = x^3$ is shown in Fig.1.48.

5. Reciprocal Function

A function $f: \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{1}{x}$ is called a reciprocal function (Fig.1.49).

6. Constant Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = c$, for all $x \in \mathbb{R}$ is called a constant function (Fig.1.50).

Understanding Special Functions in Mathematics: Constant, Identity, and Real-Valued Functions

Special Cases of Function

Special cases of function

There are some special cases of a function which will be very useful. We discuss some of them below:

• (i) Constant function
• (ii) Identity function
• (iii) Real – valued function

(i) Constant function

A function \(f : A \to B\) is called a constant function if the range of \(f\) contains only one element. That is, \(f(x) = c\), for all \(x \in A\) and for some fixed \(c \in B\).

Illustration 16

From Fig.1.37, \(A = \{a,b,c,d\}\) and \(B = \{1, 2, 3\}\) and \(f = \{(a, 3),(b, 3),(c, 3),(d, 3)\}\). Since, \(f(x) = 3\) for every \(x \in A\), Range of \(f = \{3\}\), \(f\) is a constant function.

(ii) Identity function

Let \(A\) be a non–empty set. Then the function \(f: A \to A\) defined by \(f(x) = x\) for all \(x \in A\) is called an identity function on \(A\) and is denoted by \(I_A\).

Illustration 17

If \(A = \{a,b,c\}\) then \(f=I_A = \{(a,a),(b,b),(c,c)\}\) is an identity function on \(A\).

(iii) Real valued function

A function \(f: A \to B\) is called a real valued function if the range of \(f\) is a subset of the set of all real numbers \(\mathbb{R}\). That is, \(f(A) \subseteq \mathbb{R}\).

Example 1.17

Let \(f\) be a function from \(\mathbb{R}\) to \(\mathbb{R}\) defined by \(f(x) = 3x - 5\). Find the values of \(a\) and \(b\) given that \((a, 4)\) and \((1, b)\) belong to \(f\).

Solution

\(f(x) = 3x – 5\) can be written as \(f = \{(x, 3x – 5) | x \in \mathbb{R}\}\)

\((a, 4)\) means the image of \(a\) is 4. That is, \(f(a) = 4\)

\(3a – 5 = 4 \implies a = 3\)

\((1, b)\) means the image of 1 is \(b\).

That is, \(f(1) = b \implies 3(1) – 5 = b \implies b = -2\)

Example 1.18

The distance \(S\) (in kms) travelled by a particle in time ‘\(t\)’ hours is given by \(S(t) = \frac{t^2 + t}{2}\). Find the distance travelled by the particle after

(i) three and half hours.

(ii) eight hours and fifteen minutes.

Solution

The distance travelled by the particle in time \(t\) hours is given by \(S(t) = \frac{t^2 + t}{2}\).

(i) \(t = 3.5\) hours. Therefore, the distance travelled in 3.5 hours is 7.875 kms.

(ii) \(t = 8.25\) hours. Therefore, the distance travelled in 8.25 hours is 38.16 kms, approximately.

Example 1.19

If the function \(f : \mathbb{R} \to \mathbb{R}\) is defined by:

then find the values of:

(i) \(f(4)\)

(ii) \(f(-2)\)

(iii) \(f(4) + 2f(1)\)

(iv) \(\frac{f(1) + 3f(4)}{f(-3)}\)

Solution

The function \(f\) is defined by three values in intervals I, II, III as shown by the side:

For a given value of \(x = a\), find out the interval at which the point \(a\) is located, there after find \(f(a)\) using the particular value defined in that interval.

(i) First, we see that, \(x = 4\) lie in the third interval.
Therefore, \(f(x) = 3x - 2\); \(f(4) = 3(4) – 2 = 10\)

(ii) \(x = -2\) lies in the second interval.
Therefore, \(f(x) = x^2 – 2\); \(f(-2) = (-2)^2 – 2 = 2\)

(iii) From (i), \(f(4) = 10\).
To find \(f(1)\), first we see that \(x = 1\) lies in the second interval.
Therefore, \(f(x) = x^2 – 2 \implies f(1) = 1^2 – 2 = -1\)
So, \(f(4) + 2f(1) = 10 + 2(-1) = 8\)

(iv) We know that \(f(1) = -1\) and \(f(4) = 10\).
For finding \(f(-3)\), we see that \(x = -3\), lies in the first interval.
Therefore, \(f(x) = 2x + 7\); thus, \(f(-3) = 2(-3) + 7 = 1\)

Hence,