Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f-g)(2)$$

Step-by-Step Solution

Answer

The result of \((f-g)(2)\) is 7.

Step 1: Understand the Operation

The operation \((f-g)(x)\) represents the pointwise subtraction of \(g(x)\) from \(f(x)\), which means \((f-g)(x) = f(x) - g(x)\).

Step 2: Substitute the Functions

Substitute \(f(x)\) and \(g(x)\) into the equation: \((f-g)(x) = (x^2 + 3x) - (2x - 1)\).

Step 3: Simplify the Expression

Simplify \((x^2 + 3x) - (2x - 1)\) to get \(x^2 + x + 1\). The subtraction distributes over the terms.

Step 4: Plug in the given value

Substitute \(x = 2\) into the expression \(x^2 + x + 1\).

Step 5: Calculate

Calculate \(2^2 + 2 + 1\) which simplifies to \(4 + 2 + 1 = 7\).

Key Concepts

Function Composition

Function composition involves creating a new function by combining two existing functions. This means taking the output of one function and using it as the input for another. If you have functions \(f(x)\) and \(g(x)\), their composition is denoted as \((f \circ g)(x)\), which means \(f(g(x))\). This is different from simply adding or subtracting functions.

• For function composition, always substitute the inner function completely into the outer function.
• Ensure to simplify the resulting expression step by step to avoid errors.

Function composition requires a keen understanding of how functions operate. It is not merely arithmetic but involves working through layers of calculation. This requires properly handling expressions and carefully substituting one function into another.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, and multiplication. Understanding these components is crucial when dealing with function operations. In this exercise, we handle the expressions \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\).

• Identify each component such as terms, coefficients, and constants within the expression.
• Operations like distribution and combining like terms are essential.

When substituting these expressions into operations, it's important to maintain the integrity of each term. Especially in subtraction or addition of functions, each term's proper incorporation matters significantly for a correct resolution.

Subtraction of Functions

Subtraction of functions involves taking one function and subtracting every term of another function from it. In the operation \((f-g)(x)\), each term of \(g(x)\) is subtracted from the corresponding term of \(f(x)\). This can be expressed as : \[ (f-g)(x) = f(x) - g(x) \] In our example: \[ (f-g)(x) = (x^2 + 3x) - (2x - 1) \] Let's break it down:

• First, distribute the subtraction across \(g(x)\). This means changing the signs of all terms in \(g(x)\).
• Simplify by combining like terms. For example, \(3x - 2x\) becomes \(x\).

When you subtract the functions and simplify, the outcome is a new algebraic expression. It's important to double-check each distribution step and combination to ensure accuracy. Finally, inserting specific values into the now simplified function gives you precise results, as when calculating \((f-g)(2)\). By following these steps, function subtraction becomes an efficient and clear process.