Functions
Introduction:
Functions provide us with a convenient way to handle the relationship between the values of one variable quantity that depends on the values of another variable quantity. For example, let us image a spherical rubber balloon into which air is being pumped. The radius o the balloon (r) is changing with time t. in mathematics, we say that r is a function of time t and symbolically, it may be written,
r=f(t): r is function of time t.
Similarly the volume of the balloon also depends on time t. Hence we can write
r=g(t): r is function of time t.
Different letters f and g are used because they represent different mathematical functions.
In general, if the values of a variable y depend on the values of another variable x, we write
y=f(x) i.e., y is a function of x.
Note: f(x) will obviously represent some expression of x.
Independent and Dependent Variables:
In the function y=f(x) is an expression in terms of x. Here x is known as the independent variable and y (whose values depend on x) is known as dependent variable.
Domain and Range:
While defining real-valued functions, we have to observe some restriction. One such restriction is that we can never divide by zero (0). Hence in the function
, x cannot be equal to 1.
The set of values which x can take (so that the function is well defined) is known as the set of domain for that function.
Similarly the dependent variable y in the function y=f(x), cannot take real values always. For example, in the function y=sinx, the variable y cannot take values which are greater than 1 or less than -1.
The set of values which y can take is known as the Range of the function y=f(x)
Intervals and Notations:
The domain and the range for different functions are the subsets of real numbers which are called intervals.
Open Interval:
If x can take values which lie strictly between a and b, then we can write
Closed Interval:
If x can take values which lie strictly between a and b or equal to a or equal to b, then we can write
Half-open Interval:
If only one end point is included for values of x, then the interval is called as half-open interval
Infinite Interval:
Important Functions:
- Absolute value function (modulus function)
f(x)=|x| = magnitude of x or the positive value of x.
The expression |x| can be further split as follows:
Continuity
The graph of y=|x| is continuous (i.e. no break in the curve) but has a corner at origin as shown.
Domain & Range
The Domain of the function f(x) is x∈R and Range is
- Greatest integer function (unit step function):
y=[x]= the greatest integer less than or equal to x.
It can also be simplified as:
y=[x]=n if n≤x < n+1 where n is an integer.
Continuity
The graph of f(x) is discontinuous (i.e. break in the curve) at integral value of x.
Domain & Range
The Domain of the function f(x) is x∈R and Range is y∈R.
- Signum Function:
This can also be written as
Continuity
The graph of f(x) is continuous for all values of x except at x=0 where there is a break in the curve.
Domain & Range
The Domain of the function f(x) is x∈R and Range is set {-1, 0, 1}
- Logarithmic Function:
If a>1, y increases as x increases (as seen from graph).
If 0<a<1, decreases as x increases.
Continuity
The graph of f(x) is continuous (i.e. break in the curve) in the respective domain.
Domain & Range
The Domain of the function f(x) is x>R and Range is y∈R.
Properties of logarithmic function:
- Exponential Function:
This function is the inverse of logarithmic function i.e. it can be obtained by interchanging x by y in
As observed from the graph, if a>1 then y increases as x increases.
If 0<a<1, then y decreases as x increases.
Continuity
The graph of f(x) is continuous (i.e. no break in the curve) everywhere.
Domain & Range
The Domain of the function f(x) is x∈R and Range is y>0.
- Trigonometric Functions:
Period
Period of
Period of
Continuity
The graph of is continuous (i.e. no break in the curve) everywhere.
The graph of is discontinuous (i.e. break in the curve) at
Domain & Range
The Domain of is x∈R and Range is set [-A, A].
The Domain of and Range is y∈R.
- Inverse Trigonometric Functions:
As observed from graph,
increases as x increases,
and decreases as x increases
and increases as x increases.
Continuity
The graphs are continuous (i.e. no break in the curve) everywhere in their respective domains.
Domain & Range
f(x)=sin-1x: The Domain of f(x) is set [-1, 1] and the Range is set [-π/2, π/2].
f(x)=cos-1x: The Domain of f(x) is set [-1, 1] and the Range is set [0, π].
f(x)=tan-1x: The Domain of f(x) is set (-∞, ∞) and the Range is set (-π/2, π/2).
Odd, Even and Periodic Functions:
- Even Function
If a function f(x) is symmetrical about Y-axis i.e. f(-x)=f(x) for all values of x, then the function y=f(x) is called an even function.
- Odd Function
If a function f(x) is symmetrical about origin i.e. f(x)=-f(-x) for all values of x, then the function y=f(x) is called an odd function.
- Periodic Function
If there exists a constant T such that f(x+T)=f(x) is satisfied by all values of x, then the function y=f(x) is called a periodic function.
The smallest positive value of T satisfying (f(x+T)=f(x) is known as the period of the function f(x).
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