Question

select all the correct answers.

which tables represent constant functions?

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 2 & -1 \\\\ \\hline 2 & 0 \\\\ \\hline 2 & 1 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 2 & -3 \\\\ \\hline 4 & -7 \\\\ \\hline 6 & -11 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 2 & 3 \\\\ \\hline 4 & 3 \\\\ \\hline 6 & 3 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 3 \\\\ \\hline 2 & 12 \\\\ \\hline 3 & 27 \\\\ \\hline \\end{array}$$

\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & -1 \\\\ \\hline 2 & -1 \\\\ \\hline 3 & -1 \\\\ \\hline \\end{array}$$

Explanation:

Define a constant function

A constant function is a function whose output value \(y\) is the same for every input value \(x\). This means that as \(x\) changes, \(y\) remains constant. We apply the Definition of a Function to ensure each input has exactly one output.

Analyze the first table

The first table has:

• \(x = 2, y = -1\)
• \(x = 2, y = 0\)
• \(x = 2, y = 1\)

This relation maps the single input \(x = 2\) to multiple different outputs. Therefore, it is not a function.

Analyze the second table

The second table has:

• \(x = 2, y = -3\)
• \(x = 4, y = -7\)
• \(x = 6, y = -11\)

The output \(y\) changes as \(x\) changes. This is a linear function, not a constant function.

Analyze the third table

The third table has:

• \(x = 2, y = 3\)
• \(x = 4, y = 3\)
• \(x = 6, y = 3\)

Each input \(x\) maps to exactly one output, and the output \(y\) is always \(3\). This represents a constant function.

Analyze the fourth table

The fourth table has:

• \(x = 1, y = 3\)
• \(x = 2, y = 12\)
• \(x = 3, y = 27\)

The output \(y\) changes as \(x\) changes. This is a quadratic function, not a constant function.

Analyze the fifth table

The fifth table has:

• \(x = 1, y = -1\)
• \(x = 2, y = -1\)
• \(x = 3, y = -1\)

Each input \(x\) maps to exactly one output, and the output \(y\) is always \(-1\). This represents a constant function.

Answer:

• (A) Table 1: x = [2, 2, 2], y = [-1, 0, 1]
• (B) Table 2: x = [2, 4, 6], y = [-3, -7, -11]
• (C) Table 3: x = [2, 4, 6], y = [3, 3, 3] (Correct answer)
• (D) Table 4: x = [1, 2, 3], y = [3, 12, 27]
• (E) Table 5: x = [1, 2, 3], y = [-1, -1, -1] (Correct answer)