Which table below represents a function?(1 point)

A table represents a function if each input (value of \(x\)) corresponds to exactly one output (value of \(y\)). Let's analyze each table:

1. Table 1:

   • \(x = 2\) corresponds to \(y = 1, 6, 8, 9\).
   • This means that the input \(2\) has multiple outputs, so this is not a function.

2. Table 2:

   • \(x = 5\) corresponds to \(y = 0\).
   • \(x = 3\) corresponds to \(y = 3, 1\).
   • The input \(3\) has multiple outputs, so this is not a function.

3. Table 3:

   • \(x = -2\) corresponds to \(y = 5\).
   • \(x = 0\) corresponds to \(y = 5\).
   • \(x = 2\) corresponds to \(y = 5\).
   • \(x = 4\) corresponds to \(y = 5\).
   • Each \(x\) value has exactly one \(y\) value, so this is a function.

4. Table 4:

   • \(x = 5\) corresponds to \(y = 1\) and \(y = 7\).
   • The input \(5\) has multiple outputs, so this is not a function.

Therefore, the only table that represents a function is Table 3.

In order for the relation below to be a function, which value is not allowed to be the missing x-value?

{(2, 8), (4, 5), (7, 3), (9, 0), (?, 6)}


(1 point)
Responses

0

7

3

1

In a function, each input \( x \) must correspond to exactly one output \( y \). In the given relation:

\[ {(2, 8), (4, 5), (7, 3), (9, 0), (?, 6)} \]

We can analyze the pairs:

• The existing \(x\) values are \(2, 4, 7, 9\).

To determine which \(x\) value cannot be used for the missing entry (denoted as \('?'\)), we need to ensure that the missing \(x\) value does not duplicate an existing one.

Among the possible answers:

• 0: This value is not currently in the set, and can be used.
• 7: This value is already assigned (to \(y = 3\)), so using \(7\) would mean that there would be two outputs for that same input, which violates the definition of a function.
• 3: Not in the set, can be used.
• 1: Not in the set, can be used.

Thus, the value that is not allowed to be the missing \(x\) value is 7.

Match each domain in the first column to the equivalent interval notation domain in the second column.(3 points)

4≤x<8

4<x<8

4<x≤8

Here's the matching of the provided domain descriptions to their equivalent interval notation:

1. \(4 \leq x < 8\): This includes \(4\) but not \(8\). The equivalent interval notation is **[4, 8)**.

2. \(4 < x < 8\): This does not include \(4\) and does not include \(8\). The equivalent interval notation is **((4, 8)**.

3. \(4 < x \leq 8\): This does not include \(4\) but includes \(8\). The equivalent interval notation is **((4, 8]**.

So the matches would be:

• \(4 \leq x < 8\) → *[4, 8)*
• \(4 < x < 8\) → *((4, 8)*
• \(4 < x \leq 8\) → *((4, 8]*

The cost for an upcoming field trip is $10 per student. The cost of the field trip, in dollars, is a function of the number of students.

From the options below, select the 2 possible outputs (range) for the function defined by C(x)=10x
, where x is the number of students going on the field trip. Select only 2 options.

(2 points)
Responses

5

54

60

92

20

35

The cost function for the field trip is given by:

\[ C(x) = 10x \]

where \( x \) is the number of students. Since the cost is calculated as $10 per student, the outputs (costs) will be multiples of 10 based on the number of students.

To find which options are possible outputs of this function, we can check which provided numbers are multiples of 10. The possible outputs from the given options can be calculated as follows:

• If \( x = 2 \): \( C(2) = 10 \times 2 = 20 \)
• If \( x = 3 \): \( C(3) = 10 \times 3 = 30 \)
• If \( x = 5 \): \( C(5) = 10 \times 5 = 50 \)
• If \( x = 6 \): \( C(6) = 10 \times 6 = 60 \)
• If \( x = 9 \): \( C(9) = 10 \times 9 = 90 \)

From the selected options, we look for outputs that can be calculated as \( C(x) \):

• 5: Not a multiple of 10
• 54: Not a multiple of 10
• 60: \(10 \times 6 = 60\) (valid output)
• 92: Not a multiple of 10
• 20: \(10 \times 2 = 20\) (valid output)
• 35: Not a multiple of 10

The two possible outputs (range) for the function \( C(x) = 10x \) are 60 and 20.

Evaluate the function f(x)=4−5x using the inputs below. Which input generates the largest output when evaluating the function?

(1 point)
Responses

f(4)

f(0)

f(1)

f(−2)

To determine which input generates the largest output for the function \( f(x) = 4 - 5x \), we can evaluate the function at each given input.

1. Evaluating \( f(4) \): \[ f(4) = 4 - 5(4) = 4 - 20 = -16 \]

2. Evaluating \( f(0) \): \[ f(0) = 4 - 5(0) = 4 - 0 = 4 \]

3. Evaluating \( f(1) \): \[ f(1) = 4 - 5(1) = 4 - 5 = -1 \]

4. Evaluating \( f(-2) \): \[ f(-2) = 4 - 5(-2) = 4 + 10 = 14 \]

Now, we can compare the outputs:

• \( f(4) = -16 \)
• \( f(0) = 4 \)
• \( f(1) = -1 \)
• \( f(-2) = 14 \)

The largest output is from \( f(-2) = 14 \).

Therefore, the input that generates the largest output is \( f(-2) \).

Which of the following equations correctly describes the function in the table using function notation?

x y
0 -2
2 4
4 10
6 16


(1 point)
Responses

f(x)=2x+4

f(x)=x−2

f(x)=2x

f(x)=3x−2

To determine which equation correctly describes the function represented in the table, let's analyze the relationship between \(x\) and \(y\):

The table is:

\[ \begin{array}{c|c} x & y \ \hline 0 & -2 \ 2 & 4 \ 4 & 10 \ 6 & 16 \ \end{array} \]

We can observe the pattern in the values of \(x\) and \(y\):

1. When \(x = 0\), \(y = -2\).
2. When \(x = 2\), \(y = 4\).
3. When \(x = 4\), \(y = 10\).
4. When \(x = 6\), \(y = 16\).

To find a formula that fits these points, let's calculate the change in \(y\) compared to the change in \(x\):

• From \(x = 0\) to \(x = 2\) (change of 2), \(y\) goes from \(-2\) to \(4\) (change of \(4 - (-2) = 6\)).
• From \(x = 2\) to \(x = 4\) (change of 2), \(y\) goes from \(4\) to \(10\) (change of \(10 - 4 = 6\)).
• From \(x = 4\) to \(x = 6\) (change of 2), \(y\) goes from \(10\) to \(16\) (change of \(16 - 10 = 6\)).

The change in \(y\) is consistently \(6\) for every change of \(2\) in \(x\), indicating a linear relationship.

Next, we can find the slope \(m\):

• \(m = \frac{\text{change in } y}{\text{change in } x} = \frac{6}{2} = 3\).

Now, we can express \(y\) in terms of \(x\) using the slope-intercept form, \(y = mx + b\), by first finding the \(y\)-intercept \(b\).

Using one of the points, say \((0, -2)\): \[ -2 = 3(0) + b \implies b = -2. \]

Thus, the equation is: \[ y = 3x - 2. \]

In function notation, we can express this as: \[ f(x) = 3x - 2. \]

Now, we can check if this equation fits the other points in the table:

• For \(x = 2\): \(f(2) = 3(2) - 2 = 6 - 2 = 4\) (correct).
• For \(x = 4\): \(f(4) = 3(4) - 2 = 12 - 2 = 10\) (correct).
• For \(x = 6\): \(f(6) = 3(6) - 2 = 18 - 2 = 16\) (correct).

Therefore, the correct equation describing the function in the table using function notation is:

\(f(x) = 3x - 2\).