Core 3 Functions

After completing this chapter you should be able to: Represent a mapping by a diagram, by an equation and by a graph Understand the terms function, domain and range Combine two or more functions to make a composite function Know the difference between a ‘one to one’ and a ‘many to one’ Know the relationship between the graph of a function and its inverse

Square the set {-1, 1, -2, 2, x} Mapping diagram Graph -1 1 -2 2 x 1 4 x² Domain Range y = x² This is an example of a many to one function

Mapping diagrams and graphs ‘add 4’ on the set {-3, 1, 4, 6,x} Mapping diagram Graph -3 1 4 6 x 1 5 9 10 x + 4 Domain Range y = x + 4 This is an example of a one to one function

A function is a special mapping Every element of the domain maps to exactly ONE element of the range One to one function many to one function Not a function

You can write a function two different ways f(x) = 2x + 1 OR f : x → 2x + 1 You can evaluate a function as well! g(3) when g(x) = 2x² + 3 g(2) = 2(3)² + 3 = 21 or find a value of x if g(x) = 53 53 = 2x² + 3 2x² = 50 x² = 25 x = ± 5

so to find the range of g(x) = 2x² + 3 sketch the graph and you can find the range of a function by sketching the graph so to find the range of g(x) = 2x² + 3 sketch the graph range of g(x) is g(x) ≥ 3

your turn exercise 2A page 14 exercise 2B page 16

How to change mappings into functions by changing the domain Lets look at y = √x. if the domain is all of the real numbers (𝑥∈ℝ) then this is not a function (negative numbers don’t get mapped anywhere) if you restrict the domain to x ≥ 0 then all the elements get mapped . we can write this function as f(x) = √x, with domain {𝑥∈ℝ, x ≥ 0 }

find the range of f(x) = 3x – 2, domain {x = 1, 2, 3, 4} the domain is discrete (integer values) so we draw a mapping diagram 1 2 3 4 1 4 7 10 Range of f(x) = 3x – 2 is {1, 4, 7, 10} f(x) is one to one

find the range of g(x) = x², domain {𝑥∈ℝ, -5≤ x ≤ 5} the domain is continuous so we plot a graph range of g(x) is 0 ≤ g(x) ≤ 25 g(x) is many to one

Check out example 5 on page 18 then do exercise 2C

Combining two or more functions you can combine two functions to make a new more complex function fg(x) means apply g first followed by f fg(x) is called a composite function if f(x) = x² and g(x) = x + 1 then fg(1) = f(1 + 1) = f(2) = 2² = 4 fg(3) = f(3 + 1) = f(4) = 4² = 16 fg(x) = f(x + 1) = (x + 1) ²

= 1 2 2𝑥+4 +4 = 1 2𝑛 𝑥 +4 = 1 𝑛𝑛(𝑥) =mnn(x) =mn²(x) And now you get the composite function and have to work out how the basic functions were combined (exciting!) The basic functions are m(x) = 1 𝑥 , n(x) = 2x + 4, p(x) = x² - 2 First lets look at 2 𝑥 +4 This is the same as 2 1 𝑥 +4 = 2m(x) + 4 = nm(x) And finally 1 4𝑥+12 = 1 2 2𝑥+4 +4 = 1 2𝑛 𝑥 +4 = 1 𝑛𝑛(𝑥) =mnn(x) =mn²(x) Now lets try 4x² + 16x +14 =(2x + 4) ² - 2 =[n(x)] ² - 2 = pn(x)

Exercise 2D page 22

And finally Inverse functions Inverse functions perform the opposite operation to the function. It takes elements of the range and maps them back into elements of the domain Inverse functions only exist for one to one functions The inverse of f(x) is written f-1(x) f(x) x y f-1(x)

x →square (x²) → multiply by 2 (2x²) → subtract 7 (2x² - 7) For many straightforward functions all you need is a flow chart to find the inverse h(x) = 2x² - 7 Order of operations is: x →square (x²) → multiply by 2 (2x²) → subtract 7 (2x² - 7) inverse operations are √( 𝑥+7 2 ) ← square root ← divide by 2 𝑥+7 2 ← add 7 x + 7 ← x giving h-1(x) = √( 𝑥+7 2 )

you can also find the inverse of a function by changing the subject you can also find the inverse of a function by changing the subject. f(x) = 3 𝑥 −1 let y = f(x) y = 3 𝑥 −1 y(x – 1) = 3 yx = 3 + y x = 3+𝑦 𝑦 therefore f-1(x) = 3+𝑥 𝑥 check f(4) = 1 , f-1(1) = 4 solution is correct n.b. to state the inverse function replace the y’s with x

worth remembering . . . . . . the range of the function is the domain of the inverse function and vice versa

further examples can be found on page 24 – 27 now try exercise 2E page 27