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

Inverse Functions

Aims: To find harder inverse functions for quadratics and

algebraic fractions.

Learning Outcomes

• Name: What is an inverse function? (continued from last lesson)

• Describe: How to limit a quadratic function so that it is one-one. How to find an inverse function for quadratic functions.

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

Continuing Our Work

• This lesson follows on from last lesson and deals with the types of problem that may be seen in Core 3 exams.

• The problems will involve…– finding domains and ranges of functions– finding inverse functions– using the relationship between the domains

and ranges of functions and their inverses.

Example 1 Algebraic Fractions

• Work out f-1(x)?

• What are the domain and range of f-1(x)?

• Sketch f-1(x)

3and,3

23:

xRxx

xxf

–10 10 20

–10

10

20

x

y

Example 2 Quadratics

• Complete the square to find the minimum point; use this to write the range of f(x).

• Use the completed square form to find

f-1(x)

• Sketch the graphs of f(x) and f-1(x)

3and,26)( 2 xRxxxxf

Example 2 Quadratics

• Complete the square to find the minimum point; use this to write the range of f(x).

• Use the completed square form to find

f-1(x)

• Sketch the graphs of f(x) and f-1(x)

4and,38)( 2 xRxxxxf

The Story So Far...(might seem familiar)

• The evil maths wizard Moldywart has been up to his old tricks again in an effort to remove his blemishes and generally do nasty stuff!

• It is up to Hairry, Rong and Hurtmyknee to reverse his wicked spells!

... and it is your job to help them!

Your Task

• To reverse his spells (functions) you must have all the right ingredients…– You must have the reverse spell (inverse

function)– You must know where the inverse spell must

be cast from and to (its domain and range)– The path that the spell must take (a sketch of

its graph)

Learning Outcomes

• Name: What is an inverse function? (continued from last lesson)

• Describe: How to limit a quadratic function so that it is one-one. How to find an inverse function for quadratic functions.

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

Learning Outcomes

• Name: An inverse function is a function that reverses a function (e.g. it turns each output back into the input that made it)

• Describe: To inverse a quadratic you must check that the domain means that it is a one-one function (this can be found from the completed square form) x≤or≥ minimum point x value.The completed square form is then made equal to y and the function is rearranged to find x in terms of y. The function in y is the inverse function.

• Explain: The domain of the function becomes the range of the inverse function and vice versa. Since domains are easier to identify than ranges this can be an effective way to identify the range of a function.