Math Functions

Sub Topics
A function is a relation between the output and the input. As the input of a function changes, the output also changes. Consider a function is f(a) = b then "a" is the input and "b" is the output. A set of all inputs is the domain and the set of all outputs is the range or the co-domain.

A Function is a relation between the domain and the range. We can represent a function f by the notation f : X → Y.
A function from a set X (domain) to a set Y (range) is a rule of correspondence that assigns to each element x in X exactly one element y in Y.

Graphs of Functions

Graphs can convey complete information about a function in a simple and visual way. Thus, graphs help in understanding functions and their behaviors.
The properties of the graphs of a function are linear, quadratic, rational, trigonometric, absolute, logarithmic, exponential and inverse. The graph of a function f is the set of all ordered pairs (x, f(x) ) where x is in the domain of f.
Let us study the graph of some functions:

Linear - Graphs of Functions:
In a linear function, there is single variable.
The General form of linear functions is f(x) = a x + b, x is a variable and a, b are constants.
Graph of y = x

Square Root - Graphs of Functions: In a square root graph the x variable is square rooted. The general form is f(x) = a $\sqrt{x - c}$ + d. Where a, b, c are constant.

Graph of y =
√x

Exponential - Graphs of Function: The basic exponential function is defined by f (x) = ax . Where a is the base such that a > 0 and a
1.
Graph of y = ax

Identifying Functions

A set of ordered pairs is called a relation. A function has a special kind of relation where only one range exists for each domain. Let f : X → Y is a function. Every element in X is associated with exactly one element of Y .

For Example:

F = {(1, 2), ( 2, 3), (4, 5), (6, 8)}

⇒ F is a function.

and if

F = {(1, 2), (2, 3), (1, 8), (4, 7), (6, 9)}

⇒ F is not a function.

Combining Functions

There are several ways to create new functions by combining two or more functions whose input and output units are compatible.
The basic operations to combine functions are addition, subtraction, multiplication, division and the composition of functions.
1. Arithmetic Combination of Functions:

(f + g)(x) = f(x) + g(x)

Example, f(x) = 2x and g(x) = x

then (f + g)(x) = f(x) + g(x) = 2x + x = 3x

Subtraction

(f - g)(x) = f(x) - g(x)

Example, f(x) = 5x and g(x) = 2x

then (f - g)(x) = f(x) - g(x) = 5x - 2x = 3x

Multiplication

(f * g)(x) = f(x) * g(x)

Example, f(x) = x2 and g(x) = x3

then (f * g)(x) = f(x) * g(x) = x2 * x3 = x5

Division

$\frac{f}{g}$ (x) = $\frac{f(x)}{g(x)}$,  g(x) $\neq$ 0.

Example, f(x) = x2 and g(x) = x

then $\frac{f}{g}$ (x) = $\frac{f(x)}{g(x)}$

= $\frac{x^2}{x}$ = $x$

2. Composition of functions:

A composition of functions is the applying of one function to another function. It can be represented in two ways.

(fog)(x) = f {g(x)}

and                 (gof)(x) = g {f(x)}

These are read as "f composed with g of x" and "g composed with f of x" respectively.

Example: If f(x) = 2x + 1 and g(x) = 3x find (fog)(x)

(fog)(x) = f {g(x)} = f(3x) = 2 * 3x + 1 = 6x + 1

Exponential Functions

The basic exponential function is defined by f (x) = ax . Where a is the base such that a > 0 and a 1. Every exponential function passes through the point (0, 1).

If the base of an exponential function is greater than 1, then the function will increase exponentially. If the base of an exponential function lies between 0 and 1, then the function will decrease exponentially.
The graph below shows the value of the base affect on the growth rate of an exponential function.

Inverse Functions

The inverse of a function gives us the value of x to get that value of y. Let f be a function with the domain D and a range R. A function g with the domain R and a range D is an inverse function for f if, for all x in D, y = f(x) if and only if x = g(y). The domain of f is the range of  $f^{-1}$ and the range of f is the domain of $f^{-1}$ .

A function $f^{-1}$ is the inverse of f if:

1. For every x in the domain of f,  $f^{-1}$(f(x)) = x.

2. For every x in the domain of  $f^{-1}$, f( $f^{-1}$(x)) = x.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions with a base a. We write logarithmic functions as loga(x). So, loga (x) = y <=> ay = x. The function increases if a > 1 and decreases if 0 < a < 1. The domain of the logarithmic function is the set of strictly positive real numbers, and the range is R.

From the definition, we conclude that:

For x > 0,   $a^{log_{a}(x)}$ = x

and

For all x,  loga(ax) = x

Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverse of each other, y = ex if x = ln y. Equations that involve exponential functions are in the form $log_a a^x$ = x.
The inverse of the exponential function y = ax is x = ay . The logarithmic function y = loga x is defined as equivalent to the exponential equation x = ay * y = loga x only  when x = ay , 0 < a < 1. It is called the logarithmic function with a base a .

Example:
Graph for f(x) = 2x , y = x, g(x) = logx.