Determine Functions from X to Y and Find Their Range
Question:
Let
$$
X=\{1,2,3,4\}
$$
and
$$
Y=\{1,5,9,11,15,16\}
$$
Determine which of the following sets are functions from \(X\) to \(Y\) and find the range of each function.
(i) $$ f_1=\{(1,1),(2,11),(3,1),(4,15)\} $$ (ii) $$ f_2=\{(1,1),(2,7),(3,5)\} $$ (iii) $$ f_3=\{(1,5),(2,9),(3,1),(4,5),(2,11)\} $$
(i) $$ f_1=\{(1,1),(2,11),(3,1),(4,15)\} $$ (ii) $$ f_2=\{(1,1),(2,7),(3,5)\} $$ (iii) $$ f_3=\{(1,5),(2,9),(3,1),(4,5),(2,11)\} $$
Solution
(i) Function \(f_1\)
In \(f_1\), every element of \(X\) has exactly one image in \(Y\).
Therefore, \(f_1\) is a function.
The range is:
$$ \{1,11,15\} $$
(ii) Function \(f_2\)
Here, the ordered pair \((2,7)\) is given, but
$$ 7\notin Y $$
Also, the element \(4\in X\) has no image.
Therefore, \(f_2\) is not a function from \(X\) to \(Y\).
(iii) Function \(f_3\)
Here, the element \(2\) has two images:
$$ 9 \text{ and } 11 $$
A function cannot assign more than one image to the same element.
Therefore, \(f_3\) is not a function.