Functions and graphs

05 min read

A function is a concept of mathematics that studies the dependence between variable quantities in the process of their change. For instance, with a change in the side of a square, the area of the square also varies. The question of how the change in the side of the square affects the area is answered by a mathematical relationship between the area of the square and the side of the square.Let the variable x take numerical values from the set D.

A function is a rule that attributes to every number x from D one definite number y where y belongs to the set of Real numbers. Here, x is called the independent variable and y is called the dependent variable.

The set D is referred to as the domain of the definition of the function and the set of all values attained by the variable y is called the range of the function.

In other words, a variable y is said to be the value of the function of a variable x in the domain of definition D if to each value of x belonging to this domain there corresponds a definite value of the variable y.

This is symbolised as y = f(x) where f denotes the rule by which y varies with x.

 

Even functions

A function is said to be even if for any value of x f(x) = f(-x)

Properties

  • The sum, difference, product and quotient of even functions is also an even function.
  • The graph of an even function is symmetrical about the y-axis.

Examples of even functions are y =x2, y = x4, y = |x|, cosΘ..... etc.

 

Odd functions

A function is said to be odd if for any value of x f(x) = -f(-x)

Properties

  • The sum and difference of odd functions are also an odd function.
  • The product and quotient of odd functions are also an even function.
  • The graph of an even function is symmetrical about the origin.

Examples of odd functions are y =x, y =x3, y= x3 + x, y = sinΘ.... etc.

Not all functions need to be even or odd. However, every function can be represented as the sum of an even function and an odd function.

 

Inverse of a function

Let there be a function y = f(x), which is defined for the domain D and has a range R. If the function y = f(x) is such that for every value of y (from the range of the function R) there corresponds one and only one value of x from the domain D, then the inverse function of y = f(x) exists and is given by x = g(y). Here it can be noticed that x becomes the dependent variable and y becomes the independent variable. Hence, this function has a domain R and a range D.

The graphs of two inverse functions when this change is used are symmetrical about the line y = x (which is the bisector of the first and the third quadrants).

 

Plotting a function

A function y = f(x) can be represented in the coordinate plane where x values are plotted in the horizontal X axis and the corresponding result of f(x) are plotted on the y-axis. Assume a function y = f(x) gives the results in the y-row for values in the x-row in the given table.

X 1 2 3 4 5 6 7 8 9 10
y=f(x) 1 3 6 5 4 1 2 4 6 7

 

The image reflects the plot of the function.

 

Shifting of graphs

The ability to visualize how graphs shift when the basic analytical expression is changed is a very important skill. The following points apply

  1. The shape of the graph y = f(x) + c will the same as that of the y = f(x) graph. The only difference would be that f(x) + c is shifted c units up(if c is positive) or down(if c is negative) on the x-y plot.
  2. The shape of the graph y = f(x ± c) will the same as that of the y = f(x) graph. The only difference would be that f(x ± c) is shifted c units left(if c is positive) or right (if c is negative) on the x-y plot.

 

Plotting a quadratic curve

A function  y = f(x) = ax2 + bx + c is a quadratic function. To plot this, we must convert the function into the form a(x+b)2 + c form. This can now be plotted using the shifting of graphs methods as discussed (f(x + a) + b shifting). The given diagram shows the graph of x2 + 4x +1 or (x+2)2 -3.


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