BASIC MATHEMATICS FOR NATURAL SCIENCES UNDERGRADUATE STUDENT TEXTBOOK

We have

\(\text{Dom}(f + g) = \text{Dom}(f-g) = \text{Dom}(fg) = \text{Dom}(f) \cap \text{Dom}(g)=\Re \cap \Re=\Re\)

\(\text{Dom}\left( \frac{f}{g} \right) = [\text{Dom}(f) \cap \text{Dom}(g)] {\left\{ x: g(x) = 0 \right\}} = \Re \left\{ \frac{5}{4} \right\}\)

There is yet another way of producing a new function from two given functions.

We have \((f \circ g)(\text{12}) = f(g(\text{12})) = f(\sqrt{\text{36}}) = f(6) = \frac{\text{36}}{\text{27}} = \frac{4}{3}\text{.}\)

\((f \circ g)(x) = f(g(x)) = f(\sqrt{3x}) = \frac{6\sqrt{3x}}{(\sqrt{3x})^{2} - 9} = \frac{6\sqrt{3x}}{3x - 9} = \frac{2\sqrt{3x}}{x - 3}\text{.}\)

The domain of \(f \circ g\) is \(\lbrack [0,3) \cup (3,\infty)\text{.}\)

We now explore the meaning of equality of two functions. Let \(f:A \rightarrow B\) and \(g:A \rightarrow B\) be two functions. Then, \(f\) and \(g\) are subsets of \(A \times B\text{.}\) Suppose \(f = g\text{.}\) Let \(x\) be any element of \(A\text{.}\) Then, \((x,f(x)) \in f = g\) and thus \((x,f(x)) \in g\text{.}\) Since \(g\) is a function and \((x,f(x)),\)\((x,g(x)) \in g\text{,}\) we must have \(f(x) = g(x)\text{.}\) Conversely, assume that \(g(x) = f(x)\) for all \(x \in A\text{.}\) Let \((x,y) \in f\text{.}\) Then, \(y = f(x) = g(x)\text{.}\) Thus, \((x,y) \in g\text{,}\) which implies that \(f \subseteq g\text{.}\) Similarly, we can show that \(g \subseteq f\text{.}\) It now follows that \(f = g\text{.}\) Thus two functions \(f:A \rightarrow B\) and \(g:A \rightarrow B\) are equal if and only if \(f(x) = g(x)\) for all \(x \in A\text{.}\) In general we have the following definition.