CIE Centre A-level Pure Maths P1 Chapter 3 CIE Centre A-level Pure Maths © Adam Gibson

Domain and Range We take one number, x and make another number, f(x) e.g. Suppose What is f(7)? The complete definition of a function must include a domain:

What is and is not a function Consider the following two equations Which one is a function? and One x, one y One x, two y

The range is the set of possible values for f(x) What is the range of f(x) = +√x? range domain

COMPOSITE functions g f Recall from page 197, that a function is a MAPPING from x to y To make a composite function, we first apply a function f to x, and then another function g. We call this the composite function f ( g ( x ) ) Example: g f

! IMPORTANT COMPOSITE functions 2 -15 -1 1/898 4 27 -15/2 Check your understanding. Find the value of these expressions: ! IMPORTANT -15 -1 1/898 4 27 -15/2 Remember that: means do g first, then f

Review – functions - basics Solve as many questions as possible from Exercises 9.1 and 9.2 p. 203-210 Hand in at least 20 completed solutions to me.

INVERSE functions Refer to page 211-213, and the box at the bottom of page 213. You need to remember three things:. Only a certain type of function has an inverse There is a standard process to find an inverse which you need to learn The graph of a function and its inverse have a very important pattern or relationship

INVERSE functions 2 Only a certain type of function has an inverse Are f and g one-one? ANSWER: g is not one-one because you cannot choose ONE x for each y

A function has an inverse if and only if it is one-one. INVERSE functions 3 Can we modify g to make it one-one? ANSWER: Yes. We just change the domain to be one side of the vertex. Now each value of y has only one value of x. A function has an inverse if and only if it is one-one.

INVERSE functions 4 There is a standard process to find an inverse which you need to learn If y = f (x), then we write the inverse as x = f -1 ( y ) Study this Example: Apply the method: “make x the subject”

INVERSE functions 5 ANSWERS:

INVERSE functions 6 The graph of a function and its inverse have a very important pattern or relationship The graph of a function and its inverse are symmetrical about the line y=x

INVERSE functions 7 Here is another example of inverse function symmetry. This one is very important! “Complete the square” Check the domain and range

INVERSE functions 8 Here is the process in full for a quadratic function: Why? one-one! Complete the square Make x the subject Choose the root (+ or -) – domain or range?

Odd and even An odd function has An even function has Very useful for sketching graphs!

Quadratic functions … are functions of the form: How many roots does this function have? Tasks Write down the general solution to the equation f(x)=0 Explain how knowing the values of a, b and c can help you to sketch the graph of f(x) Sketch the graph of

Another example: the modulus function |x| Defined as: What is the domain of |x|? What is the range of |x|? Is |x| odd or even? y x

Finding the range Sketch a graph of each of these functions, and then find their RANGE: (It’s very important that you learn how to do cases like k(x) )