What are functions?

• Functions are a formal way of writing mathematical expressions
  • eg.?f(x) = 3x + 2?would be a linear function
  • eg2.?g : x?? x²?+ 3x – 5?would be a quadratic function

   

What is the domain of a function?

• The?domain?of a function is the values of?x?(the “input”) are allowed to take
• For some functions,?x?cannot be certain values
  • eg.?f(x) = 1 / xx?cannot take the value 0

   

• Other times we may?choose?to restrict the values of?x
  • eg. The function?g(x) = 5 – 2x²?is used to model the height of water throughout the day where?x?indicates timeIt may make sense to limit?x?so it only covers a 24 hour period

   

• Inequalities?are used to describe the values?x?can take
• Any?exclusions?are usually indicated using the “not equal to” symbol (≠)

What is the range of a function?

• The?range?of a function is the values of?f?(the “output”) that could occur
• Some functions can never take certain values, regardless of the value of x
  • eg.?f(x) = x²f, a squared (real) value,?cannot?be negative
  • eg2.?g(x) = 1 / xg?can never be zero (because numerator cannot be 0)

   

• The?range?of a function can be influenced by its?domain
• As with the domain,?inequalities?are used to describe the values a function can take and “not equal to” (≠) is used for any exclusions

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How do I solve problems involving the domain and range

• You need to be able to?identify?and?explain?any exclusions in the domain of a function
• You need to be able to?deduce?the?range?of a function from its?expression?and?domain
• You may also be asked to sketch a graph of a function
  • This could involve sketching?parts?of familiar graphs that are restricted because of the domain and exclusions

   

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Domain and range of inverse functions

• Make sure you are familiar with?inverse?functions, denoted by?f^-1,?g^-1, etc.
• The?range?of a function is the?domain?of the inverse function
• The?domain?of a function is the?range?of the inverse function