# Functions

## Functions Synopsis

**Synopsis**

**Function**

A relation ‘f’ from a non-empty set A to another non-empty set B is said to be a**function**if every element of A has a unique image in B.

**Domain, co-domain and range of a function**

• The domain of ‘f’ is the set (of all inputs) A. No two distinct ordered pairs in ‘f’ have the same first element.

• Every function is a relation but the converse is not true.

• If f is a function from A to B and (a, b) ε f, then f(a) = b, where ‘b’ is called the**image**of ‘a’ under f and ‘a’ is called the**pre-image**of ‘b’ under f.

• If f: A → B, then A is the domain and B is the co-domain of f.

• The range of the function is the set of all images (outputs).**Real function**

• A function which has either R or one of its subsets as its range is called the real valued function.

• Moreover, a**real function**has the set of real numbers or one of its subsets both as its domain and as its range.**Types of Functions**

•**One-one (Injective) function**

A function f: X → Y is said to be one-one or injective function if an element of A is associated to an unique element of set B.

i.e. f: X → Y is one-one if and only if, for all x_{1}, x_{2}∈ X, f(x_{1}) = f(x_{2}) ⇒ x_{1}= x_{2}

•**Many-one function**

A function f: X → Y is said to be many-one if two or more elements of X have the same image in Y.

•**Onto (Surjective) Function**

A function f : A → B is an onto function, if for each b ε B, there is at least one a ε A such that f(a) = b, i.e. if every element in B is the image of some element in A, then f is an onto or surjective function.

For an onto function, range = co-domain.

•**Bijective Function**

A function which is both one-one and onto is called a**bijective**function or a bijection.

A one-one function defined from a finite set to itself is always onto, but if the set is infinite, then it is not the case.

•**Into Function**

The mapping f : A → B is called an into function if there exists at least one element in B having no pre-image in A.

•**Identity function**

i. Function f: X → X is an identity function if f(x) = x for each x ε A.

ii. Graph of the identity function is a straight line which makes an angle of 45° with both the x–axis and the y--axis, respectively. All points on this line have their x and y coordinates equal.

•**Constant function**

i. A constant function is one which maps each element of the domain to a constant i.e. for

f: R → R, f(x) = c for every x ε R.

ii. Domain of this function is R and range is the singleton set {c}, where c is a constant.

iii. Graph of a constant function is a line parallel to the x-axis. The graph lies above the x-axis if the constant c > 0, below the x-axis if the constant c < 0 and is the same as the x-axis if c = 0.

•**Polynomial function**

i. A function f: R → R defined as y = f(x) = a_{0}+ a_{1}x + a_{2}x^{2}+ …. + a_{n}x^{n}, where n is a non-negative integer and a_{0}, a_{1}, a_{2}, …a_{n}ε R is called a polynomial function.

ii. A linear polynomial function represents a straight line, while a quadratic polynomial function represents a parabola.

•**Rational function**

i. Functions of the form , where f(x) and g(x) ≠ 0 are polynomial functions, are called rational functions.

ii. Domain of rational functions does not include those points where g(x) = 0. For example, the domain of is R – {2}.

•**Modulus function**

i. The function f: R → R denoted by f(x) = |x| for each x ε R is called modulus function.

ii. The modulus function is defined as f(x) = x if x ≥ 0 and f(x) = −x if x < 0.

iii. The graph of a modulus function is above the x-axis as shown in the figure.

•**Step or greatest integer function**

i. A function f: R → R denoted by f(x) = [x], x ε R, where [x] represents the value of greatest integer, less than or equal to ‘x’ is called a step or greatest integer function. It is also called a**floor function**.

•**Smallest integer function**

A function f: R → R denoted by f(x) = [x], x ε R where is the value of the smallest integer, greater than or equal to ‘x’ is called a smallest integer function. It is also known as the**ceiling function**.

•**Signum function**

i. A function defined by f(x) = , x ≠ 0 and 0 for x = 0 is called signum function.

ii. The domain of a signum function is R and its range is {−1, 0, 1}.

•**Exponential function**

If ‘a’ is a positive real number other than unity, then a function which relates each x ε R to a^{x}is called the exponential function.

•**Logarithmic function**

If a > 0 and a ≠ 1, then the function defined by f(x) = log_{a}x, x > 0 is called the logarithmic function.

•**Reciprocal function**

The function defined by f : R - {0} → R such that, is called the reciprocal function.

•**Square root function**

The function defined by f : R^{+}→ R such that, is called the square root function.

•**Square function**

The function defined by f : R → R such that, f(x) = x^{2}is called the square function.

•**Cube function**

The function defined by f : R → R such that, f(x) = x^{3}is called the cube function.

•**Cube root function**

The function defined by f : R → R such that, is called the cube root function.

•**Inverse of a function**

i. If f: A → B is a bijective function, then for every x ∈ A, we have a y ∈ B, such that y = f (x). A new function**f**from B to A which associates each element y ∈ B to its pre-image x = f^{–1}^{–1}(y) ∈ A can be defined and this function**f**is called the inverse of function f.^{–1}

ii. Method to find f–1 if f is bijective

Step 1: Put y = f (x).

Step 2: Solve y = f (x) and express x in terms of y.

Step 3: The value of x obtained in step 2 gives f^{–1}(y).

Step 4: Replace y by x in f^{–1}(y) to obtain f^{–1}(x).

•**Even and odd functions**

i. If f (x) is a function of x such that f(–x) = f(x) for every x ∈ domain, then f (x) is an even function of x.

ii. If f (x) is a function of x such that f(–x) = – f(x) for every x ∈ domain, then f (x) is an odd function of x.**Algebra of Real functions**

Let f : X → R and g: X → R be two real functions, where X ⊂ R. Then

• (f + g): X → R is defined as (f + g) (x) = f(x) + g(x), x ε X.

• (f – g): X → R is defined as (f – g) (x) = f(x) – g(x), x ε X.

• (f ∙ g): X → R is defined as (f ∙ g) (x) = f(x) ∙ g(x), x ε X.

• (kf): X → R is defined as (kf) (x) = k (f(x)), x ε X, where k is a real number.

•

• (f + k) (x) = f(x) + k

• f^{n}(x) = [f(x)]^{n}**Composition of functions**

Let f : A → B and g : B → C be two functions. Then a function gof : A → C defined by (gof) (x) = g (f (x), for all x ∈ A is called the composition of f and g.

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