Problem 36 - Thinking Mathematically

Question

Evaluate each function at the given value of the variable. \(f(x)=4 x-3\) a. \(f(7)\) b. \(f(0)\)

Step-by-Step Solution

Answer

The solutions are \(f(7) = 25\) and \(f(0) = -3\).

Step 1: Evaluate the function at \(x=7\)

To find \(f(7)\), take the function \(f(x)=4x-3\) and substitute \(x=7\) into it. This results in \(f(7) = 4(7) - 3 = 28 - 3 = 25\).

Step 2: Evaluate the function at \(x=0\)

To find \(f(0)\), take the function \(f(x)=4x-3\) and substitute \(x=0\) into it. This results in \(f(0) = 4(0) - 3 = 0 - 3 = -3\).

Key Concepts

Substitution

Substitution is a fundamental technique in mathematics that simplifies complex expressions by replacing variables with specific values. In the context of evaluating functions, it involves substituting the given value of the variable into the function to calculate the result.
The process of substitution is straightforward:

• Identify the variable in the function, here it's often represented as \(x\).
• Replace the variable in the function with the given numerical value.
• Perform the arithmetic operations to find the function's value at that specific point.

For example, given \(f(x) = 4x - 3\), to find \(f(7)\), you substitute \(x = 7\) into the function:
\(f(7) = 4(7) - 3 = 28 - 3 = 25\).
Substitution simplifies understanding by converting abstract algebraic expressions into concrete numbers, making analysis easier.

Linear Functions

Linear functions are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They graph as straight lines and have a standard form of \(f(x) = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
Linear functions have a constant rate of change, represented by the slope \(m\). This means for every increase in \(x\), \(f(x)\) increases or decreases by the same amount.
For example, in \(f(x) = 4x - 3\), the coefficient of \(x\) is 4, indicating the function rises by 4 units for each unit increase in \(x\).

• The slope \(m = 4\) shows the steepness and direction of the line.
• The y-intercept \(b = -3\) is where the function crosses the y-axis, meaning \(f(0) = -3\).

With these attributes, linear functions are some of the simplest to analyze and graph, providing a clear understanding of relationships between variables.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make calculations more manageable and solve equations. This skill is crucial in evaluating functions as it allows us to handle terms and operations systematically.
When evaluating a function, we often need to perform several algebraic manipulations, such as:

• Substitution: Replacing variables with specific numbers, as shown in evaluating \(f(x) = 4x - 3\) at \(x = 7\) and \(x = 0\).
• Simplification: Carrying out operations like addition, subtraction, multiplication, and division to transform expressions.
• Order of operations: Ensuring calculations are done in the correct sequence, following the rules of BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction).

In the function \(f(x) = 4x - 3\), for \(f(7)\), you perform multiplication first \(4 \times 7\), followed by subtraction \(28 - 3\), resulting in 25. This methodical approach to handling algebraic expressions is vital for correctly evaluating and understanding mathematical relationships.