Functions and Graphs Index

Implicit and explicit functions

In mathematics, The explicit function is a function in which the dependent variable has been given “explicitly” in terms of the independent variable. Or it is a function in which the dependent variable is expressed in terms of some independent variables. It is denoted by: y=f(x) Examples of Explicit functions are: y=axn+bx where a , n and b are constant. y=5x3-3 The Implicit function is a function in which the dependent variable has not been given “explicitly” in terms of the independent variable. Or it is a function...

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Basic properties of Logarithms.

Logarithms are important throughout mathematics and science. Establishing some basic properties or relations of Logarithms helps us for faster computation of mathematical problems. Some of the basic properties of Logarithms and proof of them are given below, they are also called theorems of logarithm. 1>  For any positive x ,y and a: Proof: Suppose: then, and so, thus 2> For any positive a and x: Proof: suppose, then and thus,  3> For any positive x , y and a: proof: 4> For any positive a,b and x: Proof: suppose , then...

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The Logarithmic Function.

The Logarithm. If “x” is a number and, Then , “y” is known as the logarithm of “x” to the base “a”. For example: :-The logarithm of 16 to the base 2 is 4 which can be shown as: :-The logarithm of 8 to the base 2 is 3 which can be shown as: The logarithmic Function: The function in which the relationbetween input(let x) and and output(let y) is given by:  , where “a” is a constant, is known as the logarithmic function. when, , “y” is known as the logarithmic function...

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Exponential Function.

Exponential Function.

Exponential Function: Exponential function is the function which is defined by the following formula: y=f(x)=ax Where , a is a constant great than 0 and a , x both are real numbers. In any exponential function defined by formula y=f(x)=ax, “a” is said to be the base of exponential function “f” and “x” is the exponent of “a”. Some Examples of Exponential Function: a> if f:A→B is defined by f(x)=2x then “f” is exponential function of base 2. We can show this exponential function...

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Exponent(index) and laws of exponents(indices).

Exponent or Index: When a number (let a) is multiplied by itself multiple time(let n times) then we represent this case by adding “n” superscription to “a” as given below: an When , a number is repeatedly multiplied by itself the number of times the number is multiplied by itself is called the exponent the the number which is being multiplied is called base or index. For example: In an “a” the index or base and “n” is exponent of the base “a”. Laws of Exponents or Indices: The rules which gives...

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The Trigonometric Functions.

The Trigonometric Functions.

Trigonometric Functions: If we place an angle in standard position or at origin and draw a circle with center at origin such that the circle will intersect terminal arm of the angle , As shown in following figure: Where “x” is the x-coordinate of the point “P”, “y” is the y-coordinate of point “P” and “r” the radius of circle. Then for any angle of Θ there are six trigonometric functions named: Sine , Cosine , Tangent , Cosecant , Secant and Cotangent The above functions are...

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Angle and it’s measurement.

Angle and it’s measurement.

Angle: IF a line is rotated on one of it’s end point without changing the position of it’s another end point or it is rotated on one of it’s end point the the configuration formed by initial arm , terminal arm and vertex or common end point is known as Angle. An angle always has following things: An initial arm. A terminal arm.  and A vertex. As shown in figure below: An angle is said to be positive if the initial arm is rotated anti-clockwise direction and vice versa. If the vertex of any angle is at the origin of a...

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Some simple algebraic functions.

Some simple algebraic functions.

The following types of functions are known as algebraic function: Algebraic functions are also known as Polynomial functions. a. The Identity Function: The function defined by y=f(x)=x is known as identity function. For example: If I:R→R is a  function defined by f(x)=x=y then the function “I” is a identity function. We can show a identity function in graph as: b. The Constant Function: The function defined by y=f(x)=k , where “k” is a constant is known as constant function. For example: If A={1,2,3} and b={1}...

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Composite Function.

Composite Function.

What is Composite Function? Let f:A→B and g:B→C be any two functions , we can represent this two functions in a diagram as following. What is happening here is the output of function “f” is again processed by another function “g” to give a new output. If we make a new function (let the new function be “h”) which maps input of the function “f” directly to output of function “g” or maps “A” directly to “C” then the new function “h” is said...

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Inverse Function.

Inverse Function.

Inverse of a Function: Let “f” be a Function from set A to B or , elements of set A are changed or outputted to elements of set B if they are processed through or inputed in the function “f”. or f:A→B Then , inverse of the function “f” is a new  rule or function (Let the new function be function “g”)  , such that if we input the output of function “f” into the new function “g” then it produces or outputs the input of function “f” or, inverse of a...

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