Verified Answer :

  • Substitute the given \(x\)-values into the function \(f(x) = x - 2\) to calculate the corresponding outputs: \(f(-8) = -10\), \(f(-5) = -7\), and \(f(-4) = -6\).
  • Organize these results into a table of input-output pairs \((x, f(x))\): \((-8, -10)\), \((-5, -7)\), and \((-4, -6)\).
  • Plot these points on a Cartesian plane and draw a straight line passing through them.
  • Final values are \(f(-8) = -10\), \(f(-5) = -7\), \(f(-4) = -6\). The function is represented by the line passing through these points: \(\boxed{f(-8)=-10, f(-5)=-7, f(-4)=-6}\).

Explanation

  1. Problem Analysis

We are given the linear function \(f(x) = x - 2\). Our goal is to compute the outputs of this function for three specific inputs: \(x = -8, -5,\) and \(-4\), complete a corresponding table, and determine how to graph the function. A linear function of the form \(f(x) = mx + b\) represents a straight line on the Cartesian plane, where \(m\) is the slope and \(b\) is the \(y\)-intercept. Here, the slope is \(1\) and the \(y\)-intercept is \(-2\).

  1. Completing the Table

To fill in the table, we substitute each value of \(x\) into the function \(f(x) = x - 2\):

  1. For \(x = -8\): \(f(-8) = -8 - 2 = -10\).
  2. For \(x = -5\): \(f(-5) = -5 - 2 = -7\).
  3. For \(x = -4\): \(f(-4) = -4 - 2 = -6\).

The completed table is as follows:

\multicolumn{2}{|c|}{\(f(x) = x - 2\)} \) \(x\)\(f(x)\) \) -8-10 \) -5-7 \) -4-6 \)
  1. Graphing the Function

To graph the function \(f(x) = x - 2\), we treat the values in our table as coordinate pairs \((x, f(x))\). Specifically, we have the points \((-8, -10)\), \((-5, -7)\), and \((-4, -6)\).

  1. Place these points on a Cartesian coordinate plane.
  2. Since the function is linear, these points will be collinear. Use a straightedge to draw a line passing through all three points.
  3. Because the domain of a linear function is typically all real numbers unless otherwise specified, extend the line in both directions to represent the full extent of the function.

Examples

Linear functions are used in many real-life situations, such as predicting costs. For example, if a taxi service charges a base fee of \(\$2\) and a rate of \(\$1\) per kilometer, the total cost \(C\) for a ride of \(x\) kilometers is given by the linear function \(C(x) = 1x + 2\). By calculating values for different distances, you can estimate your travel budget. This illustrates how graphing linear relationships allows us to visualize costs and make informed decisions.