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Algebra and Functions

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If we have two sets, elements of one set is associated with the element of other set, known as Functions. Algebra and functions are very important for finding the relations in the given sets. Algebra is a branch of mathematics which uses letters in place of numbers. Algebra is useful in computations similar to that of arithmetic with non-numerical mathematical objects.

Algebraic Function

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Algebraic function is a function that can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication and division and raising to a fractional power. An algebraic function in one variable x is a function y = f(x) that satisfies a polynomial equation.
$a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+.....+a_{0}(x)$ = 0
where, the coefficients $a_{i}$(X) are polynomial functions of x.

They are used to calculate the value of unknown variables in the equations. To represent unknown quantity in algebra, we have to assume variable which is a letter or symbol. And, the group of numbers, symbols, real numbers and polynomials that express an operation is known as an algebraic expression.

Solved Example

Question: Find the value of  p in 5 + 9 + p + 12 = 0.
Solution:
5 + 9 + p + 12 = 0
14 + p + 12 = 0
p = -26

Types of Algebraic Functions

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Given below are the different types of algebraic functions.

Even and Odd Functions: Even and Odd functions are based on the symmetry relations, that they tend to satisfy when plotted on a graph. In mathematical analysis, like power series and fourier series, even and odd functions play a very important role.

A function f(x) is said to be even if f(-x) = f(x). The function should have a symmetry about y axis. Graph remains unchanged after reflection about the y-axis.
Examples: $x^{2}$, $x^{4}$, $x^{6}$, cos(x) etc.

A function f(x) is said to be odd if f(-x) = - f(x). An odd function has symmetry about the origin of the coordinate system.
Examples: $x^{3}$, $x^{5}$, sin(x) etc.

Linear, Quadratic and Cubic Functions:
The generalized form of a linear function will have one as its highest power.
f(x) = ax + b, where a and b are constants, and a is not equal to 0. While graphing linear function, plot each ordered pair on a coordinate plane. Connect the points with a straight line.

The generalized form of a quadratic function will have two as its highest power.
f(x) = ax$^{2}$ + bx + c, where a, b and c are constants, and a is not equal to 0. Graph of a quadratic function is a parabola.

The generalized form of a cubic function will have three as its highest power.
f(x) = ax$^{3}$ + bx$^2$ + cx + d, where a, b, c and d are constants, and a is not equal to 0.
All cubic functions have graphs with the same basic shape.

Monotonic Functions: Monotonic function is a function between ordered sets that preserves the given order. They are the functions that tend to move in only one direction as x increases. A monotonic increasing function always increases as x increases. It is always positive.
i.e. f(a) > f(b) $\forall$ a > b.
A monotonic decreasing function always decreases as x decreases.
i.e. f(a) < f(b) $\forall$ a > b. It is always negative.

Domain and Range

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Domain - The domain of a function is the complete set of possible values of the independent variable. f(x) will be the set of real numbers, where f is the function (name of the function) and 'x' is the variable.

Range - The range of a function is the complete set of all possible resulting values of the dependent variable. After substituting the values of x, we get the resulting (y) values. A function's domain and range will be in real numbers and a function easily relates an input to output.

Composition of Functions

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Composition of functions is the process of combining functions in a manner, where the output of the one function becomes the input for the next function.
For example, consider two functions g and f. Let function f be given by f(x) = x$^{5}$ and let g be the function given by g(x) = x + 8. Then, the resulting composition is gof is given as
gof(x) = g(f(x)).

g(f(x)) = x$^{5}$ + 8

One to One Function

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A function f from x to y is said to be one - one if f(x) = f(y), $\Rightarrow$ x = y, where no element of y is the image of more than one element in x. One - One functions are also known as injective.

One-One Function

One-One Function

Not One-One Function

Not One-One Function

Onto Function

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A function f from x to y is called onto if $\forall$ y in Y, there is an x in X, such that f(x) = y. All elements in Y will be used.
Onto functions are also known as surjective.

Onto Function

Onto Function

Not Onto Function

Not Onto Function

Algebraic Functions Examples

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Given below are some of the examples on algebraic functions.

Solved Examples

Question 1: Given the functions f(x) = 4x and g(x) = x$^{2}$ + 2, find (g o f)(x) and (f o g) (x)
Solution:
Given f(x) = 4x
g(x) = x$^{2}$ + 2

(g o f)(x) = g(f(x))
= g(4x)
= (4x)$^{2}$ + 2
= 16x$^{2}$ + 2

(f o g )(x) = f(g(x))
= f(x$^{2}$ + 2)
= 4(x$^{2}$ + 2)
= 4x$^{2}$ + 8

Question 2: Check whether f(x) = x$^{5}$ - 24x$^{3}$ + 12x$^{2}$ + 10 is an even or odd function.
Solution:
Given: f(x) = x$^{5}$ - 24x$^{3}$ + 12x$^{2}$ + 10
f(-x) = (-x)$^{5}$ - 24(-x)$^{3}$ + 12(-x)$^{2}$ + 10
= -x$^{5}$ + 24x$^{3}$ + 12x$^{2}$ + 10

As f(x) $\neq$ f(-x), it is not an even function
Therefore, f(x) is an odd function.

Question 3: Check whether g(x) = 5x + 8 is one-one function or onto function.
Solution:
Let us first check for one - one function.
If f(x) = f(y) $\rightarrow$ x = y
5x + 8 = 5y + 8
5x = 5y
x = y
Therefore, g(x) is one - one function and there is no need to check for onto function.