Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=5 x+6 ; x=-5$$

Step-by-Step Solution

Answer

The value of \(f(-5)\) is \(-19\).

Step 1: Substitute x in the Function

Start by substituting the value of \(x = -5\) into the function \(f(x) = 5x + 6\). This means we'll replace every occurrence of \(x\) in the function with \(-5\). This gives us: \[f(-5) = 5(-5) + 6\]

Step 2: Multiply and Simplify

The next step is to perform the multiplication. Calculate \(5 \times (-5)\), which equals \(-25\). Now, simplify the expression: \[f(-5) = -25 + 6\]

Step 3: Compute the Final Result

Finally, add \(-25\) and \(6\) together. This results in: \[f(-5) = -19\]

Key Concepts

Linear Functions

Linear functions are a fundamental concept in algebra. They are defined by equations of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The variable \(x\) is independent and the outcome \(f(x)\) is dependent on \(x\). These equations graph as straight lines, hence the name "linear."
- **The slope** (\(m\)): This constant is the rate at which \(f(x)\) changes with respect to \(x\). A positive slope means the line ascends, while a negative slope makes it descend.
- **The y-intercept** (\(b\)): This constant represents where the line crosses the y-axis (when \(x = 0\)). It directly affects the position of the line on the graph but not its direction or steepness.
Linear functions are essential as they open doors to understanding more complex functions and relationships in mathematics. Being able to identify and use linear functions is a necessary skill in many mathematical applications.

Substitution Method

The substitution method is a straightforward technique used to evaluate functions, especially when you're tasked with finding the function value for a particular \(x\). In the given exercise, substitution helps us determine \(f(-5)\) when \(f(x) = 5x + 6\).
- **Step 1**: Locate \(x\) in the function equation. In this case, it was \(f(x) = 5x + 6\).
- **Step 2**: Replace each occurrence of \(x\) in the function with the given value, \(-5\). This creates a new equation, \(f(-5)\).
The substitution method is beneficial because it simplifies the process of finding the value of a function for a specific \(x\). It allows you to work directly with numerical values instead of general formulas.

Algebraic Simplification

Algebraic simplification is the process of breaking down and reorganizing expressions to make them easier to solve or interpret. In our exercise, once we substitute \(x = -5\) into the function, the next steps involve simplifying the expression to find a numeric value.
- **Step 1**: Perform operations such as multiplication. Here, multiplying \(5\) by \(-5\) gives \(-25\).
- **Step 2**: Simplify further by performing any additions or subtractions. Adding \(-25\) and \(6\) resulted in \(-19\).
Simplification is useful because it allows us to reach a solution cleanly and efficiently. It transforms complex expressions into manageable pieces, allowing us to focus on simple arithmetic to arrive at our final answer. Simplifying expressions is a cornerstone of algebraic problem-solving and is widely used across all levels of mathematics.