Covers the various sections relating to Mapping and Inverse functions within the Advanced Mathematics programme.
Includes notes from the textbook, as well as additional class, video and research information, diagrams and practice questions.
Applicable to all IEB Grade 12s.
Mapping and Inverse functions
Mapping, functions and non-functions
Functions and non-functions:
Function
= a relationship between 2 sets, A and B, where every element of A (the input set) is mapped
onto only one element of B (the output set).
That means: for every x value there is an unique y-value.
No two ordered pairs will have the same x-value. – x not repeated
Non-function
= a relationship between 2 sets, A and B, where an element of A (the input set) is mapped onto
more than one element of B (the output set).
That means: for every x value there can be more than one y-value.
Two ordered pairs will have the same x-value. – x repeated
Mapping:
Type 1: … to function (one)
. One-to-One mapping
. Each element of the domain is mapped onto an unique value of y.
. (x is not repeated, neither is y)
Type 2: … to function (one)
. Many-to-One mapping
. Each element of the domain is mapped onto only 1 value of y.
. (x is not repeated, but y could be)
Type 3: … to non-function (many)
. One-to-Many mapping
. An element of the domain can be mapped onto many different elements of the range.
. (x is repeated, but y is not)
Type 4: … to non-function (many)
. Many-to-Many mapping
. Many elements of the domain can be mapped onto many elements of the range.
. (x is repeated, so is y).
, Determining “to-one” or “to many”
= determine if a function or non-function ∴ the vertical line test
A vertical line has equation x =, so every point on a vertical line has the same x-value.
Process = Move a vertical line across the graph.
. In how many points does this line cut the graph?
1 point only – Function ∴ to one
More than 1 – Non function∴ to many
Determining “one-to” or “many-to”
= the horizontal line test
A horizontal line has equation y =, so every point on a horizontal line has the same y-value.
Process = Move a horizontal line across the graph.
. In how many points does this line cut the graph?
1 point only – one to
More than 1 – many to
Inverses
- Multiplicative inverses = two numbers multiplied to get 1
- Additive inverses = two numbers added to get 0
Inverse function = a reflection in the line y = x.
(x ; y) (y ; x)
The inverse of a function f, written as f -1.
NOTE: f (function); f’ (derivative); f-1 (inverse)
Finding the inverse:
1. Get the formula into standard form. (y = …)
2. Swap x and y.
3. Solve so that y is the subject of the formula.
e.g. f. y = 3x – 4
. f-1. x = 3y – 4
. 3y = x + 4
𝑥+4
. y= 3
e.g. f. y = 3x2
. f-1. x = 3y2
𝑥
. y = ± √3
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