Verified Answer :

  • Substitute the value \(x = -2\) into the function \(g(x) = -4x - 4\).
  • Multiply \(-4\) by \(-2\) to get \(8\).
  • Subtract \(4\) from \(8\) to get the final result.
  • The value of the function is \(\boxed{4}\).

Explanation

  1. Analyzing the Problem and Objective

We are given a linear function defined by \(g(x) = -4x - 4\). The goal is to evaluate this function at the specific point \(x = -2\). This process involves substituting \(-2\) for every instance of \(x\) in the given equation and then simplifying the resulting expression using standard algebraic rules for multiplication and addition.

  1. Evaluating the Function

To find \(g(-2)\), we substitute \(-2\) for \(x\) into the expression \(g(x) = -4x - 4\):
\[g(-2) = -4(-2) - 4\]
First, we perform the multiplication part of the expression:
\[-4 \times -2 = 8\]
Now substitute this back into the expression:
\[g(-2) = 8 - 4\]
Finally, we perform the subtraction:
\[8 - 4 = 4\]
Thus, the value of the function at \(x = -2\) is \(4\).

Examples

Linear functions are used in many real-world scenarios to model relationships where one quantity changes at a constant rate relative to another. For example, if you are renting a bike, the total cost might be defined by a linear function: \(C(h) = 5h + 10\), where \(h\) is the number of hours and \(\$10\) is a flat service fee. Evaluating this function for a specific number of hours allows you to calculate exactly how much the rental will cost for that period. This same logic of substitution helps in budgeting, predicting growth patterns, and solving everyday math puzzles.

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