6.2 Functions
Definition 6.5 (Function). Let \(X\) and \(Y\) be two sets, not necessarily distinct. A function \(f\) from \(X\) to \(Y\), written \(f:X\longrightarrow Y\), is a subset of \(X\times Y\) such that for each \(a\in X\) there is a unique \(b\in Y\) with \((a,b)\in f\). That unique \(b\) is called the value of \(f\) at \(a\), and is written \(b=f(a)\).
Note 6.6. The whole content of the definition is in the word unique, and it makes two demands at once. Every element of \(X\) must be sent somewhere, so nothing in the domain is left unassigned; and no element may be sent to two different places. The two diagrams above are exactly this distinction drawn out.
Note 6.7. Several words are used for the same idea, and all of them appear in textbooks. The value \(f(a)\) is also called the image of \(a\) under \(f\), and a function is also called a mapping. The notation \(f:x\longrightarrow x^{2}\) says that \(f\) sends \(x\) to \(x^{2}\); the same function is written more usually as \(f(x)=x^{2}\).
Let \(f(x)=\frac {x}{x+3},\quad x\neq -3\)
Find
- 1.
- \(f(15)\)
- 2.
- \(f\left (-\dfrac {2}{7}\right )\)
- 3.
- \(f(a+b)\)
Solution.
In each case substitute the value for \(x\) in \(f(x)=\dfrac {x}{x+3}\).
1. \[f(15)=\frac {15}{15+3}=\frac {15}{18}=\frac {5}{6}.\]
2. The denominator is \(-\frac {2}{7}+3=\frac {19}{7}\), so \[f\left (-\frac {2}{7}\right ) =\frac {-\frac {2}{7}}{\frac {19}{7}} =-\frac {2}{7}\times \frac {7}{19} =-\frac {2}{19}.\]
3. \[f(a+b)=\frac {a+b}{(a+b)+3}=\frac {a+b}{a+b+3},\] valid provided \(a+b\neq -3\).
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.