Tags

, , , , , ,

Definition

A function is an equation between two quantities x and y such that one quantity can be defined in terms of the other quantity. 

Example

    y = mx + c – eq. 1

Where x and y are two quantities, m is some number, a multiple of x and c is a constant number. 

This equation may also be written as:

    x = (y – c) / m – eq. 2

Equation eq.1 defines y in yerms of x.

Equation eq.2 defines x in yerms of y.

The function defines a line on a graph. Let m=2 and c=–3, then the equation can be written as

    y = 2x – 3

This equation can be plotted on a graph with real values for x and y

Intercept

When the value of x is set to 0, the line cuts the y-axis of the graph at y = –3. This is called the x-intercept. On the graph, the coordinates are (0, –3).

When the value of y is set to 0, the line cuts the x-axis at 3/2 or 1.5. This is called the y-intercept. On the graph, the coordinates are (3/2, 0).

A straight line can be drawn on the graph with all possible values of x and y between the two intercepts and beyond.

Slope

In the generic function above, m is the called the slope of the line represented by the equation. 

The slope can be expressed in terms of x, y and the constant c.

    m = (y – c) / x

The slope is a property of the line and is a constant for all values of x and y.

It is calculated by the formula (y1 – y2) / (x1 – x2), where (x1, y1) and (x2, y2) are two points on the line.

For the function y = 2x – 3, the slope obtained is 2 by substituting the intercept points (0, –3) and (3/2, 0) in the slope formula.

Slope of a line may also be expressed in a trigonometric form. The line on the graph subtends an angle with the x-axis. The tangent of this angle is the slope. 

If theta is the angle of the line with x-axis, then tan(theta) = opp. Side/adjacent Side.

Therefore, –3/(3/2) , which is 2 below the axis, since the negative sign implies that the angle is below the axis line in the fourth quadrant). See graph below.

Graph

Graph for the function y = 2x – 3 looks as shown below (from WolframAlpha):

20130421-170324.jpg

Domain

The domain of a function is all values of x. 

Range

The range of a function is all values of y corresponding to the values of x.

In the function y = 2x – 3, the domain and range are both real numbers. 

 

Note

Consider an equation of the form ax+by+c=0, where a,b,c are real numbers and x and y are unknown quantities. This equation can be expressed in the function form as shown below:

Dividing the equation by b, it becomes

    y = -ax/b-c/b

Where slope m = -ax/b and the constant is -c/b

Advertisements