Problem 37 - Intermediate Algebra

Question

If \(f(x)=5 x-2\), find \(f(0), f(2), f(-1)\), and \(f(-4)\).

Step-by-Step Solution

Answer

\( f(0) = -2 \), \( f(2) = 8 \), \( f(-1) = -7 \), \( f(-4) = -22 \).

Step 1: Understanding the Function

The problem provides the function \( f(x) = 5x - 2 \). The goal is to find the values of this function at specific points: \( f(0) \), \( f(2) \), \( f(-1) \), and \( f(-4) \). This means we need to substitute these values into the function to determine the outputs.

Step 2: Calculate \( f(0) \)

To find \( f(0) \), substitute \( x = 0 \) into the function: \[f(0) = 5(0) - 2 = 0 - 2 = -2\] So, \( f(0) = -2 \).

Step 3: Calculate \( f(2) \)

To determine \( f(2) \), substitute \( x = 2 \) in the function: \[f(2) = 5(2) - 2 = 10 - 2 = 8\] Thus, \( f(2) = 8 \).

Step 4: Calculate \( f(-1) \)

For \( f(-1) \), use \( x = -1 \): \[f(-1) = 5(-1) - 2 = -5 - 2 = -7\] Hence, \( f(-1) = -7 \).

Step 5: Calculate \( f(-4) \)

Now let's find \( f(-4) \) by substituting \( x = -4 \) into the function: \[f(-4) = 5(-4) - 2 = -20 - 2 = -22\] Therefore, \( f(-4) = -22 \).

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of determining the output of a given function at specific input values. For a linear function like \( f(x) = 5x - 2 \), the goal is to plug in different values of \( x \) to see how it affects the output. Function evaluation is a crucial aspect of understanding how functions behave and is often the first step in many mathematical problems involving functions.

Here’s a simple formula:

• Identify the function, for example, \( f(x) = 5x - 2 \).
• Substitute the input value in place of \( x \) in the function.
• Perform the arithmetic operation to get the corresponding output, which is \( f(x) \).

By evaluating different inputs such as \( f(0), f(2), f(-1), \) and \( f(-4) \), you paint a clearer picture of how the function behaves over various intervals.

Substitution Method

The substitution method is a technique used to solve and evaluate functions, especially in algebraic equations. It involves replacing a variable with a specific value and then simplifying the expression. This method allows you to calculate the output for each given value of \( x \) in a function.

When approaching a problem:

• Take the original function; for example, \( f(x) = 5x - 2 \).
• Decide on the \( x \) value you wish to evaluate, such as \( x = 0 \).
• Substitute \( x \) with this value in the function equation.
• Replace every instance of \( x \) in the function with the value chosen and solve.

This operation simplifies the function to produce a single numeric output, demonstrating how altering the input impacts the outcome.

Algebraic Expressions

Algebraic expressions like \( 5x - 2 \) involve variables and coefficients and are the backbone of algebra. Understanding algebraic expressions is essential as it allows us to create functions, which in turn help us model real-world situations in mathematical terms.

An algebraic expression includes:

• Variables, which are placeholders represented by letters like \( x \).
• Coefficients, which are numbers multiplying the variables, such as 5 in \( 5x \).
• Constants, which are fixed numbers, like -2 in our example.

By changing the value of the variable, you can see different outputs, making algebraic expressions a versatile tool for problem-solving. Performing operations, such as function evaluations, involves manipulating these expressions according to algebraic rules and principles.