Problem 48 - Elementary and Intermediate Algebra

Question

Find \(f(3)\) and \(f(-1) .\) See Example 4. $$ f(x)=-4 x $$

Step-by-Step Solution

Answer

\(f(3) = -12\), \(f(-1) = 4\).

Step 1: Understand the Function

We are given a linear function \(f(x) = -4x\). This means that for any input value \(x\), the output is \(-4\) times the input.

Step 2: Substitute 3 into the Function

To find \(f(3)\), substitute \(x = 3\) into the function: \(f(3) = -4(3)\).

Step 3: Calculate \(f(3)\)

Perform the multiplication: \(-4 \times 3 = -12\). Thus, \(f(3) = -12\).

Step 4: Substitute -1 into the Function

Now, find \(f(-1)\) by substituting \(x = -1\) into the function: \(f(-1) = -4(-1)\).

Step 5: Calculate \(f(-1)\)

Perform the multiplication: \(-4 \times (-1) = 4\). Thus, \(f(-1) = 4\).

Key Concepts

Function Evaluation

Function evaluation is a critical part of mathematics, especially when dealing with linear functions. At its core, evaluating a function simply means finding the output value for a particular input value. In our given example, we want to evaluate a linear function defined as \( f(x) = -4x \). To evaluate this function for a specific number like 3 or -1, you substitute these values into the function to find \( f(3) \) or \( f(-1) \), respectively. Be sure to carefully replace every occurrence of \( x \) in the function with your chosen number. With this in mind, function evaluation uses fundamental simplicity to derive an answer from a function for any chosen input.

Substitution Method

The substitution method is straightforward but powerful. When you are given a function like \( f(x) = -4x \) and asked to find \( f(3) \) or \( f(-1) \), you use the substitution method. Here’s how it works:- Identify the value you need to substitute into the function.- Replace the variable \( x \) in the equation with this identified value.For example, to find \( f(3) \), substitute 3 in place of \( x \) in the equation, turning it into \( f(3) = -4(3) \). This demonstrates how substitution acts as a bridge in carrying numerical values into functions, helping find the right results.

Multiplication in Algebra

Multiplication is a fundamental arithmetic operation, frequently used in algebra. Understanding how to multiply in algebra is key when evaluating functions like \( f(x) = -4x \). This function requires you to multiply the constant coefficient -4 by the input value you substituted for \( x \). Let’s consider these steps:- When substituting \( x = 3 \), perform the multiplication: \(-4 \times 3\). The product is -12.- When substituting \( x = -1 \), perform \(-4 \times -1\). The negatives cancel out, leaving the result as 4.It's important to remember that the signs matter. Multiplying two negative numbers results in a positive product, while a negative and a positive number produce a negative product. Recognizing this pattern is crucial for solving and understanding algebraic expressions.