Representation of functions

REPRESENTATION OF FUNCTIONS

There are four possible ways its represent a function

• Algebraically

• Numerically

• Visually

• Verbally

Each one of them has some advantages and disadvantages. Let us look at them one at a time and try to understand them.

Representation of a function‒ Algebraic

It is one of the usual representations of functions. In this, functions are explicitly represented using formulas. The functions are generally denoted by lower case alphabet letters.

Example for the functions that represented by algebraically are

1. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A = πr². With each positive number r, there is associated one value of A, and we say that A is a function of r.

2. The algebraic function y = x². The rule that connects x and y given by the above equation.

3. The trigonometric function y = cos x.

4. The exponential function y = e^x.

5. The logarithmic function y = log x.

Though one of the easy and understandable ways of representing a function, it is not always easy to get the formula of the function. For those cases, we use other

methods of representation.

Representation of a function‒ Visual

This is basically the graphical representation of functions. This is very easy to understand. The input values are marked along the x‒axis. For any input value, the corresponding output value is the vertical displacement from the x‒axis. For example at x= a, the output is equal to f(a).

The graph shows the properties of the functions. For example from Figure 1, we can directly tell:

• where the graph is increasing or decreasing

• where the rate of change is more and where it is less

• where are the extrema

Thus, graphs are very beneficial for studying the behavior of the function. One drawback though it that we can't always get the exact values of all the outputs from the graph.

Representation of a function‒ Numerical

This is basically the tabular way of representing a function. The table contains two columns; one with the dependent variable and other with the independent variable. To show with an example, let us take a function f and independent variable as x. So, the dependent variable will be equal to (x). The table is given as:

The human population P in the world depends on time t. The following table gives estimates of the population P(t) at the time t, certain years. For instance,

P(2005) = 3,650,000,000

But for each value of the time t there is a corresponding value of P, and we say

that P is a function of t.

Table 1: Table representing a function

Though we have the exact value of the outputs, we can only have a finite number of such outputs. The analysis of the function and study of its behavior hence becomes difficult.

Representation of a function‒ Verbal

The cost C of mailing a large envelope depends on the weight w of the envelope. Even though, there is no simple formula connecting C and w, the post office has a method for deciding the C when w is known,

Many times functions are described more "naturally" by one method than another.

For example,

• Rather than looking at a table of values for the population of a country based on the year, it is easier to look at a graph to quickly see the trend.

• It is more useful to represent the area of a circle as a function of its radius algebraically (A = πr²), than it is to compile a table of values.

1. PIECEWISE DEFINED FUNCTION.

A piecewise‒defined function (also called a piecewise function or a hybrid function) is a function which is defined by multiple sub‒function, each sub‒function applying to a certain interval of the main function's domain (a sub‒ domain).

Consider a function defined by

Plotting the graph of the above function,

we get

The solid dot indicates that the point (‒1,2) is included on the graph; the open dot indicates that the point (‒1,1) is excluded from the graph. Since there is a break in the graph at the point x = ‒1, this is an example for a piecewise defined function.

Suppose that if we want to find the values of f(‒2), f(‒1) and f(0), then we have to use the definition of the function with respect to the values of x. For example, to find f(‒2), we have to consider the value of f(x) = 1−x, since the value of x≤‒1. Therefore, value of f(‒2) = 1 ‒ (‒2) = 3. Similarly, the value of f(‒1)=1‒(‒1) = 2. To find the value of f(0), we have to use the value of f(x) = x², since the value of x>‒1. Therefore, f(0) = 0.

ABSOLUTE FUNCTION

The absolute value of a number a is denoted by |a| which is the distance from a to 0 on the real number line. Note that distances are always positive or zero. |a| ≥ 0 for every number a.

Example, |‒3| = 3, |3| = 3,

|3‒e| = 3‒e        [3>e]

|3‒π|=π‒3        [π > з]

In general, |a| = a, if a ≥ 0 and

|a|

[a] = ‒a, if a <0

2. PROBLEMS UNDER DOMAIN AND RANGE

Domain is the interval in which the values of x lie.

Range is the interval in which the values of f (x) lie.

Example 1.

Sketch the graph of the function f(x) = 2.0 ‒ 0.4x and find the domain of the function.

Solution:

 y = 2.0‒0.4x ... (1)

To find the x intercept, put y = 0 in (1),we get.

0=2‒0.4x

0.4x = 2

 x =2/0.4

 x=5

 The x intercept is (5,0)

To find the y intercept, put x = 0 in (1),we get

y = 2

The y intercept is (0,2)

The graph of the given function is

Domain: (‒∞, ∞)

Example 2. Sketch the graph of the function f(x) = 2x‒1 and find the domain and range of the function.

Solution:

Given f(x) = 2x‒1

Let y=2x‒1... (1)

To find the x intercept, put y = 0 in (1), we get

0=2x‒1

2x = 1⇒ x=1/2

.. the x intercept is (1/2,0)

To find the y intercept, put x = 0 in (1), we get

 y = ‒1

..the y intercept is (0,‒1).

Since the expression 2x‒1 is defined for all real numbers, the domain of f is

the set of real numbers.

Since for each real number x, there is a real number f(x). Hence the range of f is the set of real numbers.

Example 3. Sketch the graph of a function f(x) = x² and find the domain and range of the function.

Solution:

Given f(x) = x²

Let y = x² ... (1)

Since the expression x² is defined for all real numbers, the domain of f is the set of real numbers.

The range of f consists of all values of f(x).

But y = f(x) = x² ≥ 0 for all values of x.

 ∴ The range of ƒ is {y/y ≥ 0} = [0,∞)

Example 4. Sketch the function f(x) = x (absolute function)

Solution:

Given f(x) = |x|

The definition of

and therefore, the function f(x) can be defined as

Therefore the graph of ƒ coincides with the line y = x to the right of y‒axis and coincides with the line y = ‒x to the left of y‒axis and the sketch of the function is given below.

Example 5. Sketch the graph of the function f(x) =  and find the domain of the function.

Solution:

Given f(x) = x + 2 if x ≤ ‒1

Let y = x + 2 ... (1)

Which is a straight line in (‒∞, ‒1].

To find the x intercept, put y = 0 in (1), we get

0=x+2

x= ‒2

the x intercept is (‒2,0)

Since f(x) = x + 2 if x ≤ ‒1, there is no y intercept.

Also given f(x) = x² if x > ‒1

Which is a parabola with vertex (0,0) in (−1,∞).

The domain is (‒∞, ‒1] U (‒1,∞).

Therefore the domain is (‒∞, ∞).

Example 6. Sketch the graph of the function

Solution:

Given

First consider y = x², if ‒2 ≤ x ≤ 0.

Which is a Parabola in ‒2 ≤ x ≤ 0.

Next consider y = 2‒x, if 0 < x < 2   ... (1)

To find the x intercept, put y = 0 in (1), we get

0=2‒x

x=2

∴ The x intercept is (2, 0)

To find the y intercept, put x = 0 in (1), we get

y = 2

The y intercept is (0,2)

The graph of the given function is

Example 7. Find the domain and range of a function f(x) = √x.

Solution:

Since square root of negative number is not defined, the domain of f consists of all values x such that x ≥0.

.. the domain is the interval [0, ∞).

The range of f consists of all values of f(x). But f(x) = √x ≥ 0 for all values of x in the domain.

..the domain is the interval [0, ∞).

Example 8. Find the domain and range of the function (x) = √[x +2].

Solution:

Given f(x) = √[x + 2]

Since square root of negative number is not defined, the domain of ƒ consists of

all values x such that

x+2≥0 ⇒ x ≥ −2.

..the domain is the interval [‒2,∞).

The range of ƒ consists of all values of f(x). But f(x) = √[x + 2] ≥ 0 for all values of x in the domain.

.. the domain is the interval [0, ∞).

Example 9. Find the domain and range of the function (x) = √[1 ‒ x²].

Solution:

Given f(x) = √[1 − x²]

Since square root of negative number is not defined, the domain of ƒ consists of all values x such that 1 ‒ x² ≥0.

‒x²‒1⇒ x² ≤1

⇒ |x|² ≤ 1 ⇒ |x| ≤ 1

 ‒1≤ x ≤1

..the domain is the interval [‒1,1].

The value of 1‒x² vary from 0 to 1 on the given domain. Also f(x) = √[1‒x²] vary from 0 to 1 on the given domain.

The range of f is [0,1].

Example 10. Find the domain of the function f(x) = √[3‒x]‒√[2+x].

Solution:

Given f(x)= √[3‒x]‒√[2+x]

Since square root of negative number is not defined, the domain of ƒ consists of all values x such that √[3‒x]‒√[2+x] ≥ 0.

Hence 3‒x > 0 and 2 + x ≥ 0;

 ‒2≤x≤3

the domain is the interval [‒2,3].

Example 11. Find the domain and range of the function f(x)=1/x

Solution: Given f(x) = 1/x

Let y = 1/x

y=1/x is not defined at x = 0.

..the domain is the interval (x/x ≠ 0} = (‒∞, 0) U (0, ∞)

Since reciprocal of non‒zero real number is a real number,

The range of f = (‒∞, 0) U (0, ∞)

Example 12. Find the domain of the function  

Solution: Given

f(x) is undefined at x = 0 and x = 1

..the domain is the interval {x/x ≠ 0 and x ≠ 1}

= (‒∞, 0) U (0,1) U (1,∞)

Example 13. Find the domain of the function f(x) = [ 2x³‒5 ] / [ x²+x‒6 ]

Solution:

Given

f(x) is undefined at x = ‒3 and x = 2.

.. the domain is the interval {x/x ≠ ‒3 and x ≠ 2}

 = (‒∞,‒3) U (‒3, 2) U (2, ∞)

EXERCISE

1. Find the domain of the functions

2. Find the domain and sketch the graph of the functions