Problem 29 - Elementary and Intermediate Algebra

Question

Let \(f(x)=2 x-5\) and \(g(x)=x+1 .\) Find each of the following function values. $$ (f+g)(8) $$

Step-by-Step Solution

Answer

The value of \((f+g)(8)\) is 20.

Step 1: Understand the Problem

We are given two functions: \(f(x) = 2x - 5\) and \(g(x) = x + 1\). We need to find the value of \((f+g)(8)\). This means we need to add the functions \(f(x)\) and \(g(x)\), and then evaluate the result at \(x = 8\).

Step 2: Find the Expression for \( (f+g)(x) \)

The sum of the functions \(f(x)\) and \(g(x)\) is found by simply adding their expressions. So, \((f+g)(x) = f(x) + g(x) = (2x - 5) + (x + 1)\). Simplify this expression by combining like terms.

Step 3: Simplify the Expression

Simplify \((2x - 5) + (x + 1)\) by combining like terms: \(2x + x - 5 + 1 = 3x - 4\). Therefore, \((f+g)(x) = 3x - 4\).

Step 4: Evaluate \( (f+g)(8) \)

Now substitute \(x = 8\) into the expression \(3x - 4\). Calculate \(3(8) - 4\).

Step 5: Calculate the Result

Perform the calculation: \(3(8) - 4 = 24 - 4 = 20\). Therefore, \((f+g)(8) = 20\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics where we find the value of a function at a specific input. Here, we deal with two functions: \(f(x) = 2x - 5\) and \(g(x) = x + 1\). To evaluate a function like \((f + g)(8)\), the process involves finding a combined expression for \((f + g)(x)\). Once we have this combined function, we simply substitute \(x = 8\) into this expression to find the resulting value. Understanding how to properly substitute and evaluate a function helps in many areas of mathematics and is especially useful in problem-solving and calculus.

• Identify the function's rule from the expression.
• Substitute the given value into the function in place of the variable \(x\).
• Make sure to simplify the resulting expression to find the value correctly.

This exercise in function evaluation emphasizes the importance of following a systematic approach: determine, substitute, and simplify.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and operations. Our task involves the expressions \(f(x) = 2x - 5\) and \(g(x) = x + 1\). When we talk about manipulating these expressions, we're often referring to basic algebraic techniques such as combining like terms. Here’s a step-by-step approach:

• Identify terms within the expressions. The terms in \(f(x)\) are \(2x\) and \(-5\), and the terms in \(g(x)\) are \(x\) and \(+1\).
• Combine like terms from both expressions. This involves adding coefficients of similar variables together. Hence, \(2x + x = 3x\) and \(-5 + 1 = -4\).
• Write down the new expression. In this case, it becomes \(3x - 4\).

It is important to organize and manage terms correctly. This skill ensures clarity and accuracy, especially when dealing with more complex expressions.

Arithmetic Operations

Arithmetic operations like addition, subtraction, multiplication, and division are core components of algebraic manipulation. In this exercise, we focus on adding two functions, \(f(x)\) and \(g(x)\). Think of this as performing arithmetic with each component of the expressions.
First, consider each term in the functions separately. By understanding that each operation needs to be conducted within its own context:

• Add the linear terms first: \(2x + x = 3x\).
• Next, handle the constant terms: \(-5 + 1 = -4\).
• Combine these results into the simplified expression \(3x - 4\).

The arithmetic operations are the essence of creating a new, simplified function from the two given expressions. They are also fundamental to other areas such as solving equations and finding derivatives.