Functions and Graphs

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Functions and Graphs : 

Functions and Graphs

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A very significant development in mathematics was the introduction of the Cartesian Coordinate system (or x-y coordinate system), developed by Rene Descartes (1596 - 1650).

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We usually draw the graph of a function using the Cartesian Coordinate system. This system made a lot of new mathematics possible, including calculus.

Introduction To Functions : 

Introduction To Functions

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A function is a rule that relates how one quantity depends on other quantities. For example,

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In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed depends on distance travelled and time taken (c) voltage depends on current and resistance (d) test marks depend on attitude, listening in lectures and doing tutorials (among many other variables!!)

Definition of a Function : 

Definition of a Function Whenever a relationship exists between two variables (or quantities) such that for every value of the first, there is only one corresponding value of the second, then we say: the second variable is a function of the first variable.

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The first variable is the independent variable (usually x), and the second variable is the dependent variable (usually y). The independent variable and the dependent variable are real numbers.

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EX.. We know the equation for the area of a circle from primary school: A = πr2 This is a function as each value of the independent variable r gives us one value of the dependent variable A.

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In the equation y = 3x + 1, y is a function of x, since for each value of x, there is only one value of y. If we substitute x = 5, we get y = 16 and no other value. The values of y we get depend on the values chosen for x. Therefore, x is the independent variable and y is the dependent variable.

Function Notation : 

Function Notation We normally write functions as: f(x) and read this as "function f of x". We can use other letters for functions. Common ones are g(x) and h(x). But there are also ones like P(t) which could indicate power at time t.

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We often come across functions like: y = 2x2+ 5x + 3 We can write this using function notation: f(x) = 2x2 + 5x + 3 Function notation is all about substitution. The value of the function f(x) when x = a is written as f(a).

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If we have f(x) = 4x + 10, the value of f(x) for x = 3 is written: f(3) = 4 × 3 + 10 = 22 When x = 3, the value of the function f(x) is 22.



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