By definition, a high-threshold function requires a large population to make the economic endeavor profitable; a high-range function draws people from far away to purchase the good or use the service. A cancer treatment center requires a large population since a relatively small proportion of people need cancer treatment, and a high range because it is likely that people who have cancer but live far away from the treatment center would travel further to be treated for cancer. The remaining choices are considered low range and low threshold, since ice cream shops, post offices, and department stores are more ubiquitous in cities than a cancer treatment center. 

Having a high-threshold and high-range means that people will travel very far to receive the services that are provided by this particular company. Out of the options provided, people are willing to travel however far they have to, even sometimes traveling more than half-way around the world to bring their child to the best pediatric cancer-treatment center that there is. They do so because of the medical needs and medical opportunities. A family in Afghanistan might travel to Israel or to Europe to seek the medical help they need when their child is diagnosed with Cancer, or perhaps all the way to the United States.

People are not willing to travel as far for any of the other options mentioned. To go to a Walgreens Pharmacy, most people are only willing to travel within 10 miles.

To go to a coffee shop, most people in the world only have to travel 1 mile, and with coffee not being a necessity and being something you could have at home, many people do not travel at all for this service.

While employees might travel 30+ miles in order to have a good job at a technology firm, customers don't usually travel very far to receive the services provided by these companies. In fact, many people simply purchase these products online and have them shipped to their location.

Shopping malls might draw people from a bit of a distance away, but it is subject to the services provided in a particular shopping mall. For example, your typical shopping mall will only draw customers from the surrounding towns, within a 20-30 mile radius at best. However, the "Mall of America" might draw customers from over 1,000 miles away as more of a tourist destination and as a mall that offers specialized "services" such as the roller-coaster theme park within the mall. But as this again is not a necessity, people are not willing to travel as far to visit this specific mall as they are going to travel to seek services for medical needs.

Central Place Theory assumes that people will always buy what they need in the closest place possible.