How Does a Function Work?

A function works as a relationship between the input and the output.

Example of a function:

f(x) = 2x + 1

The function f is the function name and the x inside the bracket is the input.

The 2x + 1 is the output.

So f(1) = 2(1) + 1 = 3

So 1 is the input and 3 is the output.

Similarly…

f(2) = 2(2) + 1 = 5

f(3) = 2(3) + 1 = 7

etc….

You must take note of something important:

Each input must only give exactly one output.

If an input gives 2 outputs, then it is NOT a function.

Example of a non-function:

f(x) = ± √x

In this case, an input of 4 will give 2 outputs which are ±2.

Therefore it is not a function.

You can refer to the diagram below for an easy reference of function and non-function.

Types of Functions

Linear Functions (y = mx + c,  x = a,  y = b) (Linear Equations)

m is the gradient and c is the y-intercept.

Quadratic Functions of Different Forms (Quadratic Equations)

– y = ax² + bx + c  where a, b and c are constants and  a ≠ 0

If  a > 0,  the graph has a minimum point

If  a < 0,  the graph has a maximum point

The graph of ax² + bx + c has an axis of symmetry which passes through the minimum or maximum point.

– Completed the Square Form

To find the x – intercepts, let y = 0

To find the y – intercepts, let x = 0

The line of symmetry is x = p.

– Factorized Form

x‒intercepts are x = a and x = b

The line of symmetry of the graph in the form  x = (a +b)/2

Cubic Functions

Reciprocal Functions

Exponential Functions, y = a^x

Logarithmic Functions (For A Maths Students)

Note:

·  Point of intersection on x-axis: (1,0)

·  Value of x is always positive.

·  y-axis is the asymptote (a line that the graph of a function approaches but never touches).

· The graph approaches the y-axis but will never touch or intersect.

Special Case: Graph of  y = lnx

Note:

• y = lnx is the inverse of y = e^x .
• the 2 graphs are a reflection along the line y = x.

Example of Function Question with Solutions

The diagram shows the graph of y = x² – x – 12

1. The graph cuts the y-axis at C (0, k), write down the value of k.
2. The graph cuts the x-axis at A and B, find the coordinates of A and B
3. Write down the equation of the line of symmetry
4. Write down the coordinates of the minimum point of the curve.

Solutions

Last Minute Revision for O Level Math?

Check out our exam guide on other topics here!

Circle Properties: O Level Exam Preparation Guide

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NIE Trained
Math Specialist
Jimmy Maths and Grade Solution Learning Centre

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