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 \(x\):
    \begin{eqnarray*}
    f(x) &=& x^2 -1 \\
    f(2) &=& 2^2-1 \\
    f(2) &=& 3 
    \end{eqnarray*}
  • Substitute \(x=x+h\), gives \[f(x+h) = (x+h)^2-1 \]