A function is a ‘job’
Function Notation
The notation f(x) defines a function named f. This is read as “y is a function of x.” The letter x represents the input value, or independent variable. The letter y is replaced by f(x) and represents the output value, or dependent variable.
Function Notation
Function Notation f(x)
You are familiar with function notation like:
y = 5x + 3 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
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
(b) f(4)
This means that our value of x is 4. So we substitute x with 4
= 3(4) + 7
= 12 + 7
= 19
(c) f(-2)
= 3(-2) + 7
= -6 + 7
= 1
This means that our value of x is -2. So we substitute x with -2
Example
Let f(x) = 4x2 – 3, find:
(a) f(3)
= 4(3)2 – 3
= 4(9) - 3
= 36 - 3
This means that our value of x is 3. So we substitute x with 3
(b) f(-5)
This means that our value of x is -5. So we substitute x with -5
= 33
= 4(-5)2 – 3
= 4(25) - 3
= 100 - 3
= 97
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) = 3x2 + 2. 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)
Example
If g(x) = 5x - 9, then:
(a) Solve g(x) = 21
This means that our expression is equal to 21
⇒ 5x – 9 = 21
⇒ 5x = 30
⇒ x = 6
Now solve the equation!
(b) Solve g(x) = -64
⇒ 5x - 9 = -64
⇒ 5x = -55
⇒ x = -11
This means that our expression is equal to -64
Now solve the equation!
Example
If f(x) = x2 – 3x, then solve f(x) = 4.
This means that our expression is equal to 4
⇒ x2 – 3x = 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
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
Example
If h(x) = 2x + 7, then write an expression for:
(a) h(3x)
= 2(3x) + 7
= 6x + 7
This means that our value of x is 3x. So we substitute x with 3x
(c) 4h(x)
= 4(2x + 7)
This means we are multiplying h(x) by 4.
= 8x + 28
(b) 3h(x)
= 3(2x + 7)
= 6x + 21
This means we are multiplying h(x) by 3.
Example
Let f(x) = x2 + 7, then write an expression for:
(a) f(x) +2
= (x2 + 7) + 2
= x2 + 9
(b) f(x + 2)
= (x +2)2 + 7
= (x +2) (x +2) + 7
This means we add 2 to f(x).
This means that our value of x is (x + 2). So we substitute x with (x + 2)
= x2 + 4x + 4 + 7
= x2 + 4x + 11
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