Verified Answer :

  • Identify the linear function \(g(x) = -4x - 1\).
  • Solve for unknown \(x\) by rearranging the equation for given \(g(x)\): \(7 = -4x - 1 \implies x = -2\) and \(-13 = -4x - 1 \implies x = 3\).
  • Evaluate unknown \(g(x)\) by substituting the given \(x\): \(g(0) = -4(0) - 1 = -1\) and \(g(-1) = -4(-1) - 1 = 3\).
  • Completed pairs are \((-2, 7)\), \((0, -1)\), \((3, -13)\), and \((-1, 3)\).
  • Final Answer: \(\boxed{-2, -1, 3, 3}\)

Explanation

  1. Analyzing the Problem

The problem asks us to complete a table of inputs and outputs for the linear function \(g(x) = -4x - 1\). This function maps an input \(x\) to an output \(g(x)\) by multiplying the input by \(-4\) and then subtracting \(1\). To find the missing values, we will evaluate the function for given inputs or solve for \(x\) when the output \(g(x)\) is provided.

  1. Solving for Inputs \(x\) Given Outputs \(g(x)\)

We first address the rows where the output \(g(x)\) is given, requiring us to solve for the input \(x\):

  1. For \(g(x) = 7\): We set up the equation \(7 = -4x - 1\). Adding \(1\) to both sides gives \(8 = -4x\). Dividing both sides by \(-4\) yields \(x = -2\).
  2. For \(g(x) = -13\): We set up the equation \(-13 = -4x - 1\). Adding \(1\) to both sides gives \(-12 = -4x\). Dividing both sides by \(-4\) yields \(x = 3\).
  1. Evaluating Outputs \(g(x)\) Given Inputs \(x\)

Next, we address the rows where the input \(x\) is given, requiring us to evaluate the function for the corresponding output \(g(x)\):

  1. For \(x = 0\): We substitute \(x=0\) into the function to get \(g(0) = -4(0) - 1 = 0 - 1 = -1\).
  2. For \(x = -1\): We substitute \(x=-1\) into the function to get \(g(-1) = -4(-1) - 1 = 4 - 1 = 3\).
  1. Conclusion and Final Table Values

Combining these results, the completed table is as follows:

  • When \(g(x) = 7\), \(x = -2\).
  • When \(x = 0\), \(g(x) = -1\).
  • When \(g(x) = -13\), \(x = 3\).
  • When \(x = -1\), \(g(x) = 3\).

Thus, the completed table entries are \(-2\), \(-1\), \(3\), and \(3\) respectively. The final table values correspond to \(x = \boxed{-2, 3}\) and \(g(x) = \boxed{-1, 3}\).

Examples

Linear functions are foundational for modeling real-world relationships where one variable changes at a constant rate relative to another. For example, if you are filling a water tank, the function \(V(t) = 5t + 10\) could represent the volume of water \(V\) in liters at time \(t\) in minutes, where the tank starts with 10 liters and fills at a constant rate of 5 liters per minute. By calculating inputs and outputs, you can determine how much water is in the tank at any given time or calculate the exact time needed to reach a desired volume.

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