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General algebra

Function notation

  • A function is a relation that takes certain numbers as input and assigns to each a definite output number.
  • The set of all input numbers is called the domain.
  • The set of the resulting output numbers is called the range.
  • Consider what happens to a length of wire when weights are added to it. For example, the heavier the weight, the more the wire stretches.
    • We call weight the independent variable and the length the dependent variable. We say length is dependent on weight or length is a function of weight.
    • We can write: \(l = f(w)\).
  • The most commonly used notation of a function is of the form \(f (x)\).
  • For example, if \(f(x) = x^2 - 1\), find \(f(2)\), \(f(x+h)\).
    • This question means that we have to calculate the value of the function \(f(x)\) at, \(x = 2\) and \(x = x+h\).
    • To evaluate \(f(2)\) if \(f(x) = x^2 - 1\) then substitute \(x=2\) for every \(x\):
      f(x) &=& x^2 -1 \\
      f(2) &=& 2^2-1 \\
      f(2) &=& 3 
    • Substitute \(x=x+h\), gives \[f(x+h) = (x+h)^2-1 \]