Answer
a) $f=\left\{ \left( \text{Zambia, 4}\text{.2} \right),\ \left( \text{Columbia, 4}\text{.5} \right),\ \left( \text{Poland,}\ \text{3}\text{.3} \right),\ \left( \text{Italy, 3}\text{.3} \right),\ \left( \text{United States, 2}\text{.5} \right) \right\}$
b) ${{f}^{-1}}=\left\{ \left( \text{4}\text{.2,}\ \text{Zambia} \right),\ \left( \text{4}\text{.5,}\ \text{Columbia} \right),\ \left( \text{3}\text{.3,}\ \text{Poland} \right),\ \left( \text{3}\text{.3,}\ \text{Italy} \right),\ \left( \text{2}\text{.5,}\ \text{United States} \right) \right\}$.
It is not a one-to-one function.
Work Step by Step
(a)
We get from the above graph 5 countries: Zambia, Columbia, Poland, Italy, and United States.
So according to each of these countries, the average number of years that the women prefer men who are older than themselves are 4.2, 4.5, 3.3, 3.3, 2.and 5 respectively.
Ordered pairs formed out of the available data will be $\left( \text{country,}\ \text{number of years}= \right)$$\left( \text{Zambia, 4}\text{.2} \right)$$\left( \text{Columbia, 4}\text{.5} \right)$$\left( \text{Poland, 3}\text{.3} \right)$$\left( \text{Italy, 3}\text{.3} \right)$$\left( \text{United States, 2}\text{.5} \right)$
So,
$f=\left\{ \left( \text{Zambia, 4}\text{.2} \right),\ \left( \text{Columbia, 4}\text{.5} \right),\ \left( \text{Poland,}\ \text{3}\text{.3} \right),\ \left( \text{Italy, 3}\text{.3} \right),\ \left( \text{United States, 2}\text{.5} \right) \right\}$
(b)
The inverse function is obtained by the exchange of the domain and the range of f. Thus, the inverse function is obtained as $\left( \text{number of years, country} \right)$
So,
${{f}^{-1}}=\left\{ \left( \text{4}\text{.2,}\ \text{Zambia} \right),\ \left( \text{4}\text{.5,}\ \text{Columbia} \right),\ \left( \text{3}\text{.3,}\ \text{Poland} \right),\ \left( \text{3}\text{.3,}\ \text{Italy} \right),\ \left( \text{2}\text{.5,}\ \text{United States} \right) \right\}$
We know that for a function to be a one-to-one function, each value of the domain must have a unique corresponding value in the range.
But, the value 3.3 in the domain corresponds to Poland as well as Italy in the range, so it is not a one-to-one function.
Thus, the relation that is the inverse of f is as follows:
${{f}^{-1}}=\left\{ \left( \text{4}\text{.2,}\ \text{Zambia} \right),\ \left( \text{4}\text{.5,}\ \text{Columbia} \right),\ \left( \text{3}\text{.3,}\ \text{Poland} \right),\ \left( \text{3}\text{.3,}\ \text{Italy} \right),\ \left( \text{2}\text{.5,}\ \text{United States} \right) \right\}$.
It is not a one-to-one function.
