composite functions f(x)=x 2 g(x)=x+5h(x)= 1 / x find as many composite functions made using two of...

Composite Functions

f(x)=x2 g(x)=x+5 h(x)=1/x

Find as many composite functions made using two of these as you can.

4)( xxff 25)( xxfg 2

1)(

xxfh

5)( 2 xxgf 10)( xxgg 51

)( x

xgh

5

1)(

xxhg

2

1)(

xxhf xxhh )(

One-One & Inverse Functions

• Know what is a one-one function and inverse function

• Understand how you can tell if a function is a one-one function? And how to find its inverse function

• Be able to explain why this idea is important and how can we make a function that is not one-one be one-one? Explain how you know write the inverse function in function notation. Does anyone read this at all? The connection between the graphs of functions and their inverses and the corresponding connection in domain and range

Vertical Line Test

• Functions are special Relations• A relation can only be called a function if every

value of x has just one corresponding output (y value).

Function/Not a Function

• Look at these graphs and show function/not a function on your whiteboards.

2 4

–4

–2

2

x

y

–2 2

–2

2

4

x

y

2 4 6 8 10 12

–2

2

4

x

y

–6 –4 –2 2 4 6

–2

2

4

x

y

2 4 6 8 10 12 14

–4

–2

2

x

y

–6 –4 –2 2 4 6 8

–4

–2

2

x

y

Many-One Functions

• Some functions give the same output for different inputs... Can you think of a simple example? Draw one on your whiteboard.

• The function f:x|→x2 is an example of a many-one function. Since if x=4 or x=-4 the output is 16 in both cases.

• Many-One Functions can cause complications if we want to reverse them can you explain why? Aside

This is also very useful... All high

security websites rely

upon it!

One-One Functions

• One-One Functions only have one unique output for each input it is not possible for them to give the same output from two different input values.

• This is another reason that we may wish to limit the domain.

• On your whiteboards try to sketch a graph that shows a one-one function.

–2 2

–8

–6

–4

–2

2

4

6

x

y

–2 2

–8

–6

–4

–2

2

4

6

x

y

¤/2 ¤

–1

1

2

x

y

One-One Functions

• Decide which of the following are one-one if not 1:1 how can you limit the domain to make them?

43)( xxf

One-One Functions


1: 2 xxf

One-One Functions


xxxf 3: 2

One-One Functions


xxf cos)(

One-One Functions


2)( 3 xxf

One-One Functions


xxxf 3: 3

One-One Functions


1: 2 xxf

Inversing one-one Functions

• Only one-one functions have an inverse “function”.

• The inverse of a function is written as f-1(x) and is the function that reverses (undoes) a function. f-1[f(x)] leaves x.

• To reverse a one-one function the easiest way is to make it y = [instead of f(x)] then make x the subject of a formula in y.

Inversing one-one Functions

• To reverse a one-one function the easiest way is to make it y = [instead of f(x)] then make x the subject of a formula in y.

R25)( E.g. , xxxf

1andR1

32)( E.g.

x, xx

xxf

5

21 x

xf

x

xxf

2

31

Graphs of Functions and Inverses

• What is the connection between the graphs of functions and their inverses?

2 4 6

2

4

6

x

y

Equation 2: y=x²

Equation 1: y=Äx

2 4 6

2

4

6

x

y

Equation 1: y=lnx

Equation 2: y=eÌ

Graphs of Functions and Inverses

• What is the connection between the graphs of functions and their inverses?

–2 2 4

–2

2

4

x

y

Equation 1: y=2x+3

Equation 2: y=½(x–3)

2 4 6 8

2

4

6

x

y

Equation 1: y=1/x²

Equation 2: y=(1/x)̂ ½

Learning Outcomes

• Explain: The connection between the graphs of functions and their inverses and the corresponding connection in domain and range.

• The inverse function is a reflection in the line y=x• This means that the x values become the y values

and the y values become the x values.• Therefore the domain of f(x) is the range of f-1(x) and

the range of f(x) is the domain of f-1(x)• This does not mean that all functions are self

inverse!

Independent Study

B1, B2, B3, & B4 p10 (solutions p156)& Exercise E p20 (solutions p160)

composite functions f(x)=x 2 g(x)=x+5h(x)= 1 / x find as many composite functions made using two of...

Documents