Different type of functions:
One to one: Every element in the domain has one and only one image in the codomain. Every element in the codomain has one and only one preimage in the domain.
Many to one: If at least two elements in the domain have the same image in the codomain, then the function is known as many to one
Onto function: If for every element in the codomain there is at least one preimage in the domain, then we can say that Range = Codomain
Into function: If there is at least one element in the codomain that does not have a preimage in the domain, then we can say that range is a proper subset of the codomain.
Even function: f(x) is even if and only if f(−x) = f(x) for all values of x. The graph of such a function is symmetric about the y-axis.
Odd function: f(x) is odd if and only if f(−x) = −f(x) for all values of x. The graph is symmetric about the origin.
- If f(x) is an odd function and f(0) exists, then f(0) = 0
Identity function:
A function f defined by f(x) = x for all real value of x is called the identity function.
If you refer to the graph, then you will find that the identity function is nothing but a straight line that passes through the origin and is inclined at an angle of 45° with the x-axis. Algebraically, we can also represent it as y = x.
Graph of f(x) = x2
A function given by f(x) = x2 is called the square function.
Clearly, y = x2 is a parabola. It’s also an even function, so it is symmetrical about the y-axis as shown in the figure:
Graph of f(x) = x3
A function given by f(x) = x3 is called the cube function. When a function is an odd function, its graph is symmetrical about the opposite quadrant or we can say that the graph is symmetrical about the origin as shown in the figure:
Graphs of rational expressions:
- Graph of f(x) = 1/x
This is also called the reciprocal function or a rectangular hyperbola. This is also an odd function, so its graph is symmetrical about the opposite quadrant.
Graph of f(x)=1/x2
Since the power of x is even, this is clearly an even function. So, it must be symmetrical about the y-axis. Also, even negative values of the function f(x) will always be positive. It is also important to mention here that the domain of the function will have all real values except 0, i.e., it cannot take the value 0.
Piecewise functions:
There are a few types of functions that come under the category of piecewise functions. They are absolute value function, signum function, greatest integer function, fractional part function, least integral function etc.
Absolute value function (or modulus function):
- y= |x| = {x; x ≥ 0 or x; x < 0}
It is a numerical value of x.
- It is symmetrical about the y-axis.
Exponential function
The function f(x) = ax, where a ≠ 0 and x is a real number, is called exponential function.
Depending upon the value of a, we can have all the increasing functions or decreasing functions.
Case I: If a > 1
Let us take an example where f(x) = y = 5x increases with an increase in x.
Case II: If 0 < a < 1
Let us take an example where f(x) = y = 10x decreases with an increase in x.
The exponential function in general increases or decreases as (a > 1) or (0 < a < 1), respectively.
Logarithmic function
This is an inverse of an exponential function. In other words, in the exponential function, if we replace x with y and y with x and rearrange the function by taking x as the independent variable and as the dependent variable, then the new function will be a logarithmic function.
The logarithmic function is defined as f(x) = logax; where (x, a >0)
f(x) = logax; (x, a > 0) and a ≠ 0. The domain of logarithmic function consists only of positive real numbers.
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