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