Before we start working on functions, let’s practice an exam question on topics we have already studied. This is from the winter 2016 paper 11:

**Solution** **(mark scheme)**

**Functions**

A function is a relation that uniquely associates members of one set with members of another set (e.g. f(x) = 2x is a function).

Functions can be **one-one** (e.g. f(x) = 5x) or **many-one **(e.g. f(x) = 2x^{2})

**Syllabus extract showing what we must know about functions**

We can write functions using the notation f(x) = 3x or f: x |-> 3x (I prefer the first one, but you will see the second one sometimes and need to be able to understand it.

The most important concepts that must be understood about functions are those of the **domain **and the **range**.

**Domain**

The **domain** is the set of input values that the function can take.

For instance f(x) = 2x can take all values, so its domain is or (-∞,∞).

However cannot take the value 2, so its domain is or x (-∞,2)∪(2,∞).

**Range**

The **range** is the set of output values that the function can take.

For instance, with f(x) = 2x, f(x) can take all values, so its range is or (-∞,∞)

However, with f(x)= 1/(x-2), f(x) will never take the value 0, so its domain is or x (-∞,0)∪(0,∞).

**Worked Examples**

**Example solutions**

**Exercise**

Let’s complete some questions from the following exercise:

**Answers**

**Composite Functions**

We can easily combine functions by making one function the **internal function** and one function the **external function**.In this case, the variable in the external function is replaced by the internal function.

e.g. f(x) = x+2, g(x)=3x. What would fg(x) be? What would gf(x) be?

The range of the internal function is necessarily the domain of the external function.

**Worked examples**

As stated above, the **range** of the internal function is the **domain **of the external function. So if f(x)=x^{2} and g(x)=x+3, then what will be the range of gf(x)?

**Exercise**

**Answers**

**Inverse Functions**

The **inverse** of a function undoes whatever the functions does: ff^{-1}(x)=f^{-1}f(x)=x

The domain of f^{-1}(x) is the range of f(x)

The range of f^{-1}(x) is the domain of f(x)

A inverse function only exists if there is a one-one mapping (if the mapping is many-one the inverse will not be a function unless its domain is restricted).

If f(x) and f^{-1}(x) are the same function, then f is a **self-inverse** function, e.g. .

- Process for finding inverse functions:
- Write y=f(x);
- Swap the variables y and x throughout;
- Rearrange to make y the subject of the equation;
- Check the domain of the inverse function (this is the range of the original function.

**Worked Examples**

**Exercise**

**Answers**

Let’s start thinking about functions **geometrically**!

**Graphs of functions (and their inverse)**

If we know what the graph of a function looks like, how will we draw the graph of its inverse?

What do we know about the domain and the range of the inverse function?

**Worked Examples**

**Exercise**

**Answers**

**Exam Question Practice**

Maybe it’s time to practice a past paper question? (Paper 12 from summer 2016)

**Mark Scheme**

**Geometrical Transformations of Functions**

**Transformation 1: Translations**

The simplest kind of transformation is a **translation**.

A function can be translated **horizontally **(parallel to the y-axis) or **vertically **(parallel to the x-axis).

is the vector notation to refer to a horizontal translation by a and a vertical translation by b.

A vertical translation of **b**, i.e. moving a function vertically up by b, is achieved by **adding** b to the function, i.e. to y.

A horizontal translation of **a**, i.e. moving a function horizontally to the right by a, is achieved by **subtracting** a from the argument, i.e. x, of the function.

**Worked Example**

- Sketch the graph of each of these functions:
- y = f(x) +2;
- y = f(x-1);
- y = f(x+1) – 2.

**Exercise – Translating Functions**

**Answers**

**Transformation 2: Reflections**

We will consider reflecting the function in the y-axis (i.e. horizontally) and reflecting it in the x-axis.

Reflecting in the y-axis, y-coordinates stay the same and every x-coordinate is replace with its negative. This is achieved by taking **f(-x)**

Reflecting in the x-axis, x-coordinates stay the same and every y-coordinate is replace with its negative. This is achieved by taking **-f(x)**

**Worked Example**

**Exercise**

**Answers**

**Transformation 3: Stretches**

y=a.f(x) stretches the graph vertically with a scale factor of a;

If 0<a<1, a “stretch” will in reality be a compression

If a<0, the stretch will be the same as the stretch by the absolute value of a that is then reflected in the x-axis.

y=f(ax) stretches the graph horizontally with a scale factor of 1/a (i.e. this will be a compression if a>1)

**Worked Example**

**Exercise**

**Answers**

**Combining Transformations**

We have to take care about the **order** in which we perform our transformations, **especially**with horizontal transformations.

With vertical transformations we work outwards, applying the transformation to the whole function

e.g. If f(x)=x^{2} is stretched vertically by 2 and then translated up by 3, we change f(x)=x^{2} to f(x)=2x^{2}+3

Horizontal transformations are not so obvious. Here we work inwards, applying the transformation only to the **argument**, i.e. only to x.

e.g. If f(x)=x^{2} is stretched horizontally by 2 and then translated right by 3, we change f(x) to

**Worked Example – Combining Transformations**

**Exercise on Combined Transformations and Combined Exercise on Functions**

**Answers**