Question

Table:
x: 0, 1, 2, 3.
y: -5, -9, -13, -17.

Solve the equation using the table
f(?) = ? - ?

Answer 1

To solve the equation \( f(x) = y \), first, we can rewrite the available data from the table given:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -5 \\
1 & -9 \\
2 & -13 \\
3 & -17 \\
\hline
\end{array}
\]

Next, we can observe the relationship between \( x \) and \( y \). The \( y \) values decrease as \( x \) increases. To understand the relationship, let's analyze the differences:

- From \( x = 0 \) to \( x = 1 \): \( -5 \to -9 \) (difference: -4)
- From \( x = 1 \) to \( x = 2 \): \( -9 \to -13 \) (difference: -4)
- From \( x = 2 \) to \( x = 3 \): \( -13 \to -17 \) (difference: -4)

The differences are constant and equal to -4, which indicates that the function is linear.

We can thus form the equation for a linear function \( f(x) \) in the form:

\[
f(x) = mx + b
\]

where \( m \) is the slope, and \( b \) is the y-intercept. From our observations:

- The slope \( m = -4 \)
- To find \( b \), we can use one of the points. Let's use the point where \( x = 0 \): \( f(0) = -5 \). Therefore, \( b = -5 \).

Putting it together, we have:

\[
f(x) = -4x - 5
\]

Now, to solve \( f(?) = ? - ? \), we can substitute \( x \) into the equation. We can choose any value of \( x \) from the table provided.

Let's choose \( x = 1 \):

\[
f(1) = -4(1) - 5 = -4 - 5 = -9
\]

Therefore, the left side equals \( -9 \). Now we can express the right side as:

If we let \( a = 1 \) (the \( x \) value chosen) and \( b = 8 \) (which we need to solve):

\[
? - ? = 1 - 10 = -9
\]

This fits. Hence, solving the equation \( f(?) = ? - ? \):

Assuming we finalize it with \( ? = 1 \), then we expressed it as \( 1 - 10 = -9 \).

So a solution could have been:

1. \( f(1) = -9 \)
2. \( 1 - 10 = -9 \)

In conclusion, the relationship can match as demonstrated and there can be valid multiple interpretations led to the outcome described!