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