In algebra we sometimes use function notation, e.g. f(x)=2x-3. One definition of a function is a relationship that for every input value has exactly one output value.

In this situation we can find out the function value for various values of x, e.g. for x=2 or x=-5.

There is another way of writing this that will sometimes be used by Cambridge. It is f: x –> 2x-3. It means the same.

Other letters can be used instead of f, for instance, g or h.

**Example 1**

Given that f(x) = x^{2} – 3x and g(x) = 4x – 6, find the value of:

(a) f(6), (b) f(-3), (c) g(1/2), (d) g(6)

**Example 2**

Given the function h: x –> 9 – x^{2}, find the value of:

(a) h(0), (b) h(3), (c) h(9), (d) h(-9)

**Example 3**

If f(x) = 3 + 2x and f(x) = 6, find x.

**Example 4**

Given the functions f(x) = x^{2} and g(x) = x+2:

(a) Solve the equation f(x) = g(x)

(b) Solve the equation 4g(x) = g(x) – 3.

**Section 1: Substituting Values into Function**

**Exercise F1**

Note that in question 6, it should say f(x)=3 and not f(x)=3f(x)=3, which is impossible.

**Answers F1**

**Section 2: Composite Functions**

A composite function is “a function of a function”.

So if f(x) = 3x -4 and g(x) = 1-x^{2} we can make the composite functions as follows:

fg(x) = 3(1-x^{2}) – 4

gf(x) = 1 – (3x – 4)^{2}

Notice that these are not the same, and fg(3) would be 3(1 – 9) – 4 = -28, whereas gf(3) would be 1 -25 = -24.

**Example**

Given the functions f(x) = x^{2}-2x and g(x) = 3-x, find the value of:

(a) gf(4), (b) fg(4), (c) ff(-1) and (d) gg(100).

**Exercise F2**

**Answers F2**

**Section 3: Inverse Functions**

An inverse function is a function that undoes the effect of the original function. So for instance, because f(x) = 3x multiplies its inputs by 3, the inverse function is f^{-1}(x)=x/3 which divides its inputs by 3, taking them back to where they were at the start.

To find the inverse of a function, we follow the special method of swapping the variables and then rearranging the subject of the equation, which we will see in the worked examples below. It is worth being award that not all functions have an inverse.

**Worked Examples**

Find the inverse of f(x) = 3x-4

Given g(x) = 5-2x, find g^{-1}(x)

Find the inverse of the function f(x) = 3x – 4

Given g(x) = 5 – 2x, find g^{-1}(x).

**Exercise F3**

**Answers F3**