Problem 41 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$f(4)$$

Step-by-Step Solution

Answer

\(f(4) = -18\)

Step 1: Substitution

To find \(f(4)\), we need to substitute \(x = 4\) in the function \(f(x)\). Therefore, we will replace \(x\) in \(f(x) = -5x + 2\) with \(4\).

Step 2: Calculate f(4)

Now that we know we will evaluate \(f(x)\) for \(x = 4\), let's calculate the result: \(f(4) = -5(4) + 2\)

Step 3: Simplify the expression

Now, we will apply the operations in the expression to arrive at the final result: \(f(4) = -20 + 2 = -18\) Thus, \(f(4) = -18\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. In the realm of algebra, symbols such as letters stand for unknown numbers, and when these symbols are combined with different operations, they can form expressions, equations, and functions. Understanding algebra involves grasping how these symbolic forms can be used to represent and solve problems.

When working with algebraic expressions or functions, like those in the exercise, you'll often see variables being used. Variables are the letters that stand for unknown values. The goal is to understand how to manipulate these expressions by performing operations such as addition, subtraction, multiplication, and division.

Through algebra, you'll learn how to solve equations, meaning you'll find the value of the variable that makes the equation true. It's important to practice these skills, as they form the foundation for more advanced mathematics.

Function Substitution

Function substitution is a fundamental concept that involves replacing a variable within a function with a specific value or even another expression. This allows us to evaluate the function for particular numbers or inputs.

In this exercise, the task was to find the output of the function \(f(x) = -5x + 2\) when \(x = 4\). This means we need to substitute \(4\) in place of \(x\).

Steps in Function Substitution:

• Identify the function and the variable to substitute. In our case, it was \(f(x) = -5x + 2\).
• Replace the variable with the given value. Here, substitute \(x = 4\) into the equation.
• Solve the expression after substitution to find the result.

This concept is essential because it establishes the process of evaluating functions at specific points, which is a key step in analyzing and understanding their behavior.

Simplifying Expressions

Simplifying expressions is about reducing a mathematical expression to its most basic form without changing its value. This involves combining like terms, applying arithmetic operations, and following the order of operations rules (PEMDAS/BODMAS).

After substituting \(x = 4\) into the function \(f(x) = -5x + 2\), we arrived at \(f(4) = -5(4) + 2\). The next step was simplifying this expression:

1. Multiplying \(-5\) and \(4\) gives \(-20\).
2. Adding \(-20\) and \(2\) gives \(-18\).

The order of operations ensures that you correctly evaluate expressions, and it teaches how to methodically break down problems into manageable parts. By simplifying expressions, you make equations easier to understand and solve.