Problem 3 - Algebra 1: Concepts and Skills

Question

Evaluate the function \(f(x)=-5 x-2\) for the given value of \(x\) $$ x=4 $$

Step-by-Step Solution

Answer

After evaluating the function for the given \(x\), the answer is -22.

Step 1: Identify the Function and the Given Value of X

The function as given is \(f(x)=-5 x-2\) and the value of \(x\) provided is 4.

Step 2: Substitute X into the Function

This involves replacing \(x\) in the function with the given value of \(x=4\). The function, thus becomes \(f(4)=-5(4)-2\).

Step 3: Perform Arithmetic Operations

Now, we perform the arithmetic operations. Multiply -5 with 4 and subtract 2 from the result, i.e., \(-5(4)=-20\) and then \(-20-2=-22\).

Key Concepts

Function Evaluation

When we talk about function evaluation, we are referring to the process of determining the output of a function given a specific input. A function, in the broadest sense, is like a machine that takes an input (usually denoted as 'x') and produces an output after applying a set of rules. These rules are represented as mathematical operations that are to be performed on the input.

For example, if we have a function represented as f(x) and we want to evaluate this function for a particular value of 'x', say 4, our primary job is to replace the variable 'x' with the number 4 in the function's equation. The evaluation process enables us to see the direct effect of the input on the output without having to graph the entire function or derive further insights. Understanding this concept is essential because it's a fundamental step in many areas of mathematics and its applications.

Substitution Method

The substitution method is an essential tool in mathematics, particularly when evaluating functions. This method involves replacing a variable in an equation with its given or known value. By substituting the known value into the equation, we can simplify the problem and find the solution.

Take our example function, f(x) = -5x - 2, when asked to find f(4), we substitute 'x' with 4 and rewrite the equation as f(4) = -5(4) - 2. It is critical to substitute correctly to avoid any errors in the evaluation. Using parentheses around the substituted value is a good practice because it helps to prevent mistakes with signs and ensures that all arithmetic operations are applied correctly to the substituted value.

Arithmetic Operations

In the context of function evaluation, arithmetic operations are the calculations we perform once the substitution is complete. The basic operations include addition, subtraction, multiplication, and division. After substituting the value of 'x' into the function, we use these operations to simplify the expression and find the result.

After substituting 4 into our function to get f(4) = -5(4) - 2, the next step is to multiply -5 by 4 to get -20. We then subtract 2 to arrive at our final answer of -22. Each step of the arithmetic process must be carried out with care to ensure accuracy. This step-by-step approach helps in reducing the risk of errors and makes complex problems manageable.