Answer the questions below to determine what kind of function is depicted in the table below.

Question

xx 00 11 22 33 44
f, of, xf(x) minus, 4−4 minus, 9−9 minus, 14−14 minus, 19−19 minus, 24−24
Answer
Attempt 1 out of 2

This function is
because

Answer 1

To determine the kind of function depicted in the table, let's first analyze the values given for \( x \) and \( f(x) \):

\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline 0 & -4 \ 1 & -9 \ 2 & -14 \ 3 & -19 \ 4 & -24 \ \hline \end{array} \]

Now, let's determine the pattern in the function values. To do this, we can calculate the difference between successive \( f(x) \) values.

\[ \begin{align*} f(1) - f(0) & = -9 - (-4) = -5 \ f(2) - f(1) & = -14 - (-9) = -5 \ f(3) - f(2) & = -19 - (-14) = -5 \ f(4) - f(3) & = -24 - (-19) = -5 \ \end{align*} \]

We see that the difference between successive values of \( f(x) \) is constant and equal to \(-5\). This indicates that the function is linear because a linear function has a constant difference.

Thus, we can conclude that:

This function is linear because the differences between the output values \( f(x) \) are constant (specifically, each value decreases by 5 as \( x \) increases by 1).

The general form of a linear function can be expressed as \( f(x) = mx + b \). The constant difference (-5) represents the slope (\( m \)), and we can further identify the function's equation if needed. In this case, it can be represented as \( f(x) = -5x - 4 \).