1. Function Notation
A function is a ‘job’

2. Function Notation f(x)
You are familiar with function notation like:
y = 5x or y= x2 + 4x + 6
y = f(x) means that y is a function of x.
You read f(x) as ‘f of x’.
So, if y = x2 + 2, we can also write
f(x) = x2 + 2

3. Example If f(x) = 3x + 7, find: (a) f(1) = 3(1) + 7 = 3 + 7 = 10
This means that our value of x is 1. So we substitute x with 1
(a) f(1)
= 3(1) + 7
= 3 + 7
= 10
This means that our value of x is 4. So we substitute x with 4
(b) f(4)
= 3(4) + 7 =
= 19
This means that our value of x is -2. So we substitute x with -2
(c) f(-2)
= 3(-2) + 7 =
= 1

4. Example Let f(x) = 4x2 – 3, find: (a) f(3) = 4(3)2 – 3 = 4(9) - 3
This means that our value of x is 3. So we substitute x with 3
(a) f(3)
= 4(3)2 – 3
= 4(9) - 3 =
= 33
(b) f(-5)
= 4(-5)2 – 3
This means that our value of x is -5. So we substitute x with -5
= 4(25) - 3 =
= 97

5. Try these… (1) Let f(x) = 7x – 8. Find the value of: (a) f(2) (b) f(8)
(c) f(-8)
(2) Let f(x) = 3x Find the value of each of these.
(a) f(4)
(b) f(-1)
(c) f(22)
(3) Let g(x) = 3x2 – 2x + 1. Find:
(a) g(3)
(b) g(-2)
(c) g(0)

6. Example If g(x) = 5x - 9, then: (a) Solve g(x) = 21 ⇒ 5x – 9 = 21
This means that our expression is equal to 21
(a) Solve g(x) = 21
⇒ 5x – 9 = 21
Now solve the equation!
⇒ 5x = 30
⇒ x = 6
(b) Solve g(x) = -46
This means that our expression is equal to -46
⇒ 5x – 9 = -46
⇒ 5x = -55
Now solve the equation!
⇒ x = -11

7. This means that our expression is equal to 4
Example
If f(x) = x2 – 3x, then solve f(x) = 4.
⇒ x2 – 3x = 4
This means that our expression is equal to 4
⇒ x2 – 3x – 4 = 0
⇒ (x - 4)(x + 1) = 0
Now solve the equation!
⇒ (x - 4) = 0 or (x + 1) = 0
⇒ x = 4 or x = -1

8. Try these… (1) Let h(x) = 2x – 5. Solve h(x) = 7.
(2) Let g(x) = 4x - 3. Solve g(x) = 0.
(3) h(x) = x2 – 2
(a) Find h(3) and h(-6)
(b) Solve h(x) = 7
(4) Let f(x) = 3x2 – 11x.
(a) Find f(-3)
(b) Solve f(x) = 20

9. Example If h(x) = 2x + 7, then write an expression for: (a) h(3x)
This means that our value of x is 3x. So we substitute x with 3x
= 6x + 7
(b) 3h(x)
= 3(2x + 7)
This means we are multiplying h(x) by 3.
= 6x + 21
(c) 4h(x)
= 4(2x + 7)
This means we are multiplying h(x) by 4.
= 8x + 28

10. This means we add 2 to f(x).
Example
Let f(x) = x2 + 7, then write an expression for:
(a) f(x) +2
= (x2 + 7) + 2
This means we add 2 to f(x).
= x2 + 9
This means that our value of x is (x + 2). So we substitute x with (x + 2)
(b) f(x + 2)
= (x +2)2 + 7
= (x +2) (x +2) + 7
= x2 + 4x
= x2 + 4x + 11

11. Try these… (1) Let g(x) = 5x + 6. Write an expression for: (a) g(3x)
(b) 3g(x)
(2) h(x) = x Write an expression for each of these.
(a) 2h(x)
(b) h2(x)
(c) h(x) + 3
(d) h(x+3)
(3) Let p(x) = 7 – 3x. Write an expression for:
(a) 3p(x)
(b) p(3x)
(c) p(x) + 3
(d) p(x+3)
(4) f(x) = 3x2 - 2x. Write an expression for each of these.
(a) f(x – 1)
(b) f(x) - 1
(c) f(-x)
(d) -f(x)

12. Self Assessment
Do you understand how to use and solve problems involving function notation?
Yes – I fully understand the topic.
Mostly – I just need a little more practice.
Not really – I am finding this topic quite difficult.