Question

Use the table below to determine whether​ f(x) could represent a linear function. If it​ could, write​ f(x) in the form ​f(x)equals=mxplus+b.
x
minus−2
0
2
4
​f(x)
1111
66
11
negative 4−4
Question content area bottom
Part 1
Select the correct choice​ and, if​ necessary, fill in the answer box in your choice below.
A.
The values could represent the linear function ​f(x)equals=enter your response here.
​(Use integers or fractions for any numbers in the​ expression.)
B.
The function is not linear.

Answer 1

To determine whether \( f(x) \) could represent a linear function, we can check if the differences in \( f(x) \) values are constant as \( x \) increases.

The given data points are:

\[
\begin{align*}
x: & \quad -2, \quad 0, \quad 2, \quad 4 \\
f(x): & \quad 11, \quad 6, \quad 11, \quad -4 \\
\end{align*}
\]

**Step 1: Calculate the differences in \( f(x) \)**

Let's find the values of \( f(x) \) for the successive pairs of \( x \):

- From \( x = -2 \) to \( x = 0 \):
\[
f(0) - f(-2) = 6 - 11 = -5
\]

- From \( x = 0 \) to \( x = 2 \):
\[
f(2) - f(0) = 11 - 6 = 5
\]

- From \( x = 2 \) to \( x = 4 \):
\[
f(4) - f(2) = -4 - 11 = -15
\]

**Step 2: Analyze the differences**

The differences in \( f(x) \) are not constant:
- The change from \( x = -2 \) to \( x = 0 \) is \( -5 \).
- The change from \( x = 0 \) to \( x = 2 \) is \( 5 \).
- The change from \( x = 2 \) to \( x = 4 \) is \( -15 \).

Since the differences are not consistent, \( f(x) \) does not represent a linear function.

**Conclusion:**

B. The function is not linear.