# Mathematics

### Functions

In this tutorial, you will learn the following in detail:

• Functions
• Composite Functions
• Inverse Functions

#### Functions

y = x2 + 3x -2 is said to be a function of x - an expression of x. This can also be written, in a bit more advanced way as f(x) = x2 + 3x -2 - it is read as you write, function of x, f(x)
So, f(x) = x2 + 3x -2
Now whatever you put in as 'x' on the left, substitutes 'x' on the right.

E.g.
• f(2) = 22 + 3x2 -2 = 4 + 6 - 2 = 8
• f(-2) = (-2)2 + 3x(-2) -2 = 4 - 6 - 2 = -4
• f(0) = 02 + 3x0 -2 = 0 + 0 - 2 = -2
• f(x+1) = (x+1)2 + 3(x+1) -2 = x2 + 2x + 1 + 3x + 3 -2 = x2 + 5x + 2
• f(x-1) = (x-1)2 + 3(x-1) -2 = x2 - 2x + 1 + 3x - 3 -2 = x2 + x - 5
• f(2x) = (2x)2 + 3(2x) -2 = (2x)2 + 6x -2 = 4x2 + 6x - 2
• f(x/2) = (x/2)2 + 3(x/2) -2 = x2/4 + 3x/2 -2 = x2/4 + 3x/2 - 2
• f(x²) = (x²)² + 3(x²) - 2 = x4 + 3x² -2

#### Composite Functions

A function of a function is called a composite function.

E.g.1

f(x) = 2x-1 | g(x) = x²

fg(x) = 2(x²) -1
gf(x) = (2x-1)²
f²(x) = ff(x) = 2(2x-1) - 1 = 4x - 2 - 1 = 4x - 3
g²(x) = gg(x) = (x²)² = x4

E.g.2

f(x) = 3x-2 | g(x) = x² | h(x) = (x + 3)/2

Find fgh(x)
gh(x) = [(x+3)/2]² = (x+3)²/4
fgh(x) = ff(x) = 3(x+3)²/4 - 2

E.g.3

f(x) = 3x | g(x) = x² | h(x) = x-1

Find gfh(x)
fh(x) = 3(x-1) = 3x - 3
gfh(x) = (3x - 1)²

E.g.4

f(x) = 3x-1 | g(x) = x² - 2x

Find gf(1)
f(1) = 2
gf(1) = 1² -2(1) = -1

E.g.5

f(x) = 3x-2 | g(x) = 2x + 7

If gf(b) = 21, find b.
gf(x) = 2(3x - 2) + 7 = 6x - 4 + 7 = 6x + 3
gf(b) = 6b + 3
6b + 3 = 21
6b = 18
b = 3

#### Inverse Functions - f-1(x)

If two functions exist in such a way that the input of one of them is the output of the other or vice versa, they are a function and its own inverse.

E.g. g(x) = 2x- 3; -2 ≤ x ≤ 3

x g(x)
-2 -7
-1 -5
0 -3
1 -1
2 1
3 3

E.g. h(x) = (x + 3)/2; -7 ≤ x ≤ 3

x h(x)
-7 -2
-5 -1
-3 0
-1 1
1 2
3 3

Since the above two functions swap around their input and output, they are a pair of a function and its inverse.
If f(x) = 2x - 3, then f-1(x) = (x + 3)/2 and vice versa.

#### Turning a function into its inverse function

Method 1 Method 2

1. Swap x and y around.
2. Make y the subject.

E.g.
f(x) = 2x - 3
y = 2x - 3
Swapping x and y around,
x = 2y - 3
y = (x + 3)/2
f-1(x) = (x + 3)/2

#### Important facts about an inverse function

1. ff-1(x) = f-1f(x) = x
2. The function and its inverse are symmetrical in the y = x line.

#### Practice Questions

1. y = -3x + 2
2. y = 2 + 3(x + 1)
3. y = 3 - 2(4x + 1)
4. y = 1/(x - 2)
5. y = 3 - 2x/3
6. y = x² - 3
7. If f(x) = 2x - 7, solve f-1(x).
8. If f(x) = 3 + 2(x - 3), solve f-1(x).
9. if f(x) = 2x + 1 and g(x) = x², solve fg(x) = gf(x).
10. If f(x) = 4x - 3 and g(x) = x/3, solve f-1(x) = g-1(x).

Move the mouse over, just below this, to see the answers:

1. y = (2-x)/3
2. y = (x-2)/3 - 1
3. y = (1 - x)/8
4. y = 1/x + 2
5. y = 3(3 - x)/2
6. y = √(x+3)
7. x = 7
8. x = 3
9. x = 0 and x = -2
10. x = 0.27

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