**All About Functions: O Level Math Exam Preparation Guide**

**What are Functions?**

A function relates an input to an output. In simple words, a function is a relationship between inputs where each input is related to **exactly one** output.

It is like a machine that has an input and an output.

And the function is the machine which determines the output.

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**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 *= m*x* + 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 = ax^{2} + 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 ax^{2} + 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:__

__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 = x ^{2} – x – 12*

- The graph cuts the
*y*-axis at C (0,*k*), write down the value of*k*. - The graph cuts the
*x*-axis at A and B, find the coordinates of A and B - Write down the equation of the line of symmetry
- 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!

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To Your Child’s Success,

Mr Tan Kok Heng

NIE Trained

Math Specialist

Jimmy Maths and Grade Solution Learning Centre

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