Determine the range of the function modeled by the equation. In late August, Ellis is given some stamps by his uncle. On September 1, he begins to buy new stamps to add to his collection. The number of stamps in his collection can be determined using the equation f(x)=40x+57 , in which x indicates the number of weeks that have gone by since September 1 and f(x) represents the total number of stamps in his collection. However, Ellis loses interest in his collection after five weeks and stops purchasing new stamps, invalidating the ability of the function to describe the number of stamps he has after that point. How can the range of this real-world function be described using inequality symbols?(1 point) Responses 57≤y≤257 57 less-than-or-equal-to y less-than-or-equal-to 257 57

Question

Determine the range of the function modeled by the equation. In late August, Ellis is given some stamps by his uncle. On September 1, he begins to buy new stamps to add to his collection. The number of stamps in his collection can be determined using the equation f(x)=40x+57 , in which x indicates the number of weeks that have gone by since September 1 and f(x) represents the total number of stamps in his collection. However, Ellis loses interest in his collection after five weeks and stops purchasing new stamps, invalidating the ability of the function to describe the number of stamps he has after that point. How can the range of this real-world function be described using inequality symbols?(1 point) Responses 57≤y≤257 57 less-than-or-equal-to y less-than-or-equal-to 257 57<y<257 57 less than y less than 257 40<y<57 40 less than y less than 57 −∞<y<257

GPT-4o mini · Answer

To determine the range of the function $ f(x) = 40x + 57 $ for $ x $ in the context given, we need to consider the valid interval for $ x $. Since Ellis starts collecting stamps on September 1 and loses interest after 5 weeks, the values of $ x $ will range from 0 to 5. Now, we can calculate the minimum and maximum values of $ f(x) $ when $ x $ is in the interval [0, 5]: 1. **When $ x = 0 $**: $$ f(0) = 40(0) + 57 = 57 $$ 2. **When $ x = 5 $**: $$ f(5) = 40(5) + 57 = 200 + 57 = 257 $$ Thus, the function $ f(x) $ takes values starting from 57 when $ x = 0 $ up to 257 when $ x = 5 $. Given this, the range of $ f(x) $ can be described in inequality symbols as: $$ 57 \leq y \leq 257 $$ Therefore, the correct response is: **57 ≤ y ≤ 257**