Question

thomas starts a canned food collection with 2 cans of food. he challenges his friends to bring 4 cans on the second day, 8 cans on the third day, and 16 cans on the fourth day. thomas and his friends plan to collect cans for 10 days. which statement describes why this sequence is a function? select each correct answer. each day corresponds to a unique amount of cans in this sequence. the graph of the sequence would pass the horizontal line test. each amount of cans corresponds to a unique day in this sequence. the graph of the sequence would pass the vertical line test.

Explanation:

Step1: Recall Function Definition

A function is a relation where each input (domain element) has exactly one output (range element). For a sequence (where input is day, output is cans), we check:
- **Each day (input) → unique cans (output)?** Let's list days and cans: Day 1: 2, Day 2: 4, Day 3: 8, Day 4: 16... Each day (input) has one unique amount of cans (output). So "Each day corresponds to a unique amount of cans" is correct.
- **Each amount (output) → unique day (input)?** For example, if we had two days with same cans, this would fail. But here, cans are \(2, 4, 8, 16...\) (powers of 2), so each amount is unique to a day? Wait, no—wait, the sequence is \(a_n = 2^n\) (Day 1: \(2^1=2\), Day 2: \(2^2=4\), Day 3: \(2^3=8\), Day 4: \(2^4=16\)...). So each amount is unique to a day? Wait, but the question is about why it's a function. A function requires each input (day) has one output (cans). The "each day → unique cans" is about input → output (correct for function). "Each amount → unique day" is output → input (not required for function; a function can have multiple inputs with same output, but here it's one - to - one, but the key for function is input → output uniqueness).
- **Horizontal line test?** Horizontal line test is for inverse functions (to check if inverse is function). Not relevant for checking if a relation is a function (vertical line test is for graphs of functions).
- **Vertical line test?** The graph of the sequence (points \((1,2),(2,4),(3,8),(4,16)...\))—a vertical line (x = day) will intersect the graph at most once (since each day has one point). So "The graph of the sequence would pass the vertical line test" is correct, because vertical line test checks if a graph represents a function (each x - value has one y - value).

Wait, let's re - evaluate each option:
1. "Each day corresponds to a unique amount of cans in this sequence." → In a function, each input (day) has one output (cans). This is the definition of a function (domain element → unique range element). Correct.
2. "The graph of the sequence would pass the horizontal line test." → Horizontal line test is for checking if a function is one - to - one (invertible). A function doesn't need to pass horizontal line test (it can have multiple inputs with same output). So this is not a reason for being a function. Incorrect.
3. "Each amount of cans corresponds to a unique day in this sequence." → This is about output → input. A function allows multiple inputs to have the same output (though in this case, it's one - to - one, but this is not the definition of a function). The definition of a function is input → output uniqueness, not output → input. So this is not the reason for being a function. Incorrect.
4. "The graph of the sequence would pass the vertical line test." → The graph of the sequence (as a set of points \((n, a_n)\))—a vertical line \(x = n\) (day) will intersect the graph at exactly one point (since each day \(n\) has one \(a_n\)). So the graph passes vertical line test, which means it's a function. Correct. Wait, but earlier I thought "each day → unique cans" is also correct. Wait, let's check the original problem's options again (from the image):

Options:
- Each day corresponds to a unique amount of cans in this sequence.
- The graph of the sequence would pass the horizontal line test.
- Each amount of cans corresponds to a unique day in this sequence.
- The graph of the sequence would pass the vertical line test.

Wait, maybe I made a mistake earlier. Let's clarify:

- A function is a relation where for every input (x - value, here day), there is exactly one output (y - value, here cans). So:

- "Each day (input) corresponds to a unique amount of cans (output)" → This is the definition of a function (input → unique output). Correct.

- "The graph of the sequence would pass the vertical line test" → The vertical line test is used to determine if a graph represents a function (if any vertical line intersects the graph at more than one point, it's not a function). The graph of the sequence (points \((1,2),(2,4),(3,8),...\))—a vertical line \(x = 1\) intersects at \((1,2)\), \(x = 2\) at \((2,4)\), etc. So it passes vertical line test, meaning it's a function. Correct.

Wait, but maybe the problem considers both "Each day corresponds to a unique amount of cans" and "The graph of the sequence would pass the vertical line test" as correct? Wait, let's check the sequence:

Day 1: 2 cans, Day 2: 4 cans, Day 3: 8 cans, Day 4: 16 cans...

- For "Each day corresponds to a unique amount of cans": Yes, because each day (1,2,3,4...) has only one number of cans (2,4,8,16...). So input (day) → unique output (cans). This is the function definition.

- For "The graph of the sequence would pass the vertical line test": The graph is a set of discrete points \((n, 2^n)\). A vertical line (x = n) will hit at most one point (since each n has one \(2^n\)). So the graph passes vertical line test, which is a way to check if a graph is a function. So this is also correct.

Wait, but maybe the intended answers are "Each day corresponds to a unique amount of cans in this sequence" and "The graph of the sequence would pass the vertical line test"? Wait, no—let's re - read the options:

Option 1: "Each day corresponds to a unique amount of cans in this sequence." → Correct (function: each input has unique output).

Option 2: "The graph of the sequence would pass the horizontal line test." → Horizontal line test is for one - to - one functions (invertible). A function doesn't need to pass this. Incorrect.

Option 3: "Each amount of cans corresponds to a unique day in this sequence." → This is about output → input. A function doesn't require this (e.g., \(y = x^2\) has two x's for one y, but it's a function). So this is not a reason for being a function. Incorrect.

Option 4: "The graph of the sequence would pass the vertical line test." → The graph of the sequence (points \((1,2),(2,4),...\))—a vertical line (x = day) will intersect the graph at only one point (since each day has one can amount). So the graph passes vertical line test, which means it's a function. Correct.

So the correct options are "Each day corresponds to a unique amount of cans in this sequence" and "The graph of the sequence would pass the vertical line test". But wait, maybe the problem's sequence is a geometric sequence \(a_n=2\times2^{n - 1}=2^n\) (Day 1: 2, Day 2: 4, Day 3: 8, Day 4: 16...). So each day (n) has a unique \(a_n\), and the graph (as a set of points) passes vertical line test.

Answer:

• Each day corresponds to a unique amount of cans in this sequence.
• The graph of the sequence would pass the vertical line test.