Functions

• Understand about relations and functions

• Use function notation to describe simple functions

• A relation is a set of ordered pairs.

• Relations show the mapping of one set of data to another, eg pairing height with weight, or distance with time.

• The first elements in the ordered pairs (often labeled the x values) form the domain. The second elements (y values) form the range.

• Functions are special relations where each x value has only one y value.

• More formally, each element of the domain maps to only one element of the range.

• On a graph, any vertical line cuts the graph of a function at most once.

Which of the graphs is a function? Why?

• Using a set of ordered pairs:

  {(2, 5), (3, 7), (9, 19)}

• As an infinite set described by an equation:

  {(x, y): y = 2x + 1} (Read as ‘pairs of values x and y such that y = 2x + 1’.)

• As a mapping:

  f: x --> 2x + 1 (Read as ‘a function where x maps to 2x + 1’.)

• Using function notation:

  f(x) = 2x + 1 (Read as ‘f of x equals 2x + 1’.)

• In function notation, x is the input (the independent variable) and f(x) is the output (dependent variable).

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• If f(x) = 2x + 1, then f(7) = 2(7) + 1 = 14 + 1 = 15.

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• And so (7, 15) is a point on the function. (When the input is 7, the output is 15.)

• If f(x) = x³ – 4x,

  then  f(–2) = (­–2)³ – 4(–2)    = 0

  ​

• And so (–2, 0) is a point on the function.

  (When the input is –2, the output is 0.)

• If f(x) = 5x - 2. What is the value of x, when f(x) = 8?

  Answer:

  ​8 = 5x - 2

  5x = 10,

  x = 10/2 = 5