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) = mx + b.

Answer 1

`\( f(x) = (-7)/(2)x + 6 \)` is the linear representation of the given function.

To determine whether f(x) represents a linear function, we can check if the ratio of the differences in f(x) and x is constant. If it is, then the function is linear.

Let's calculate the differences:

`\text{Difference in } x &: 0 - (-2) = 2`

`\text{Difference in } f(x) &: 6 - 13 = -7`

Now, let's calculate the ratio:

`\[\text{Ratio} = \frac{\text{Difference in } f(x)}{\text{Difference in } x} = (-7)/(2)\]`

Since the ratio is constant,
`\( f(x) \)` represents a linear function. Now, let's find the equation in the form
`\( f(x) = mx + b \)`.

We know that
`\( m \)`, the slope, is given by the ratio. So,

`\[m = (-7)/(2)\]`

Now, let's use the point-slope form of a line
`(\( y - y_1 = m(x - x_1) \))` with one of the points from the table. Let's use the point
`\((-2, 13)\)`:

`\[f(x) - 13 = (-7)/(2)(x - (-2))\]`

Simplify this equation:

`\[f(x) - 13 = (-7)/(2)(x + 2)\]`

Now, solve for
`\( f(x) \)`:

`\[f(x) = (-7)/(2)x - 7 + 13\]`

Combine the constants:

`\[f(x) = (-7)/(2)x + 6\]`

So,
`\( f(x) = (-7)/(2)x + 6 \)` is the linear representation of the given function.