Problem 17 - A Graphical Approach to College Algebra

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

(f-g)(2) = 7.

Step 1: Understand the Problem

We need to find the value of the function \((f-g)(x)\). This means we need to perform the operation \(f(x) - g(x)\) and then evaluate it at \(x=2\).

Step 2: Write the Expression for (f-g)(x)

The expression for \((f-g)(x)\) is \(f(x) - g(x)\). We substitute the given functions to get:\[(f-g)(x) = (x^{2} + 3x) - (2x - 1).\]

Step 3: Simplify the Expression

Simplify the expression by performing the subtraction:\[(f-g)(x) = x^2 + 3x - 2x + 1 = x^2 + x + 1.\]

Step 4: Evaluate (f-g)(x) at x=2

Substitute \(x=2\) into \(x^2 + x + 1\) to find \((f-g)(2)\):\[(f-g)(2) = 2^2 + 2 + 1 = 4 + 2 + 1 = 7.\]

Key Concepts

Composition of Functions

When working with functions, knowing how to compose them is crucial. Composition essentially means plugging one function into another. If you have two functions, say, \( f(x) \) and \( g(x) \), the composition is denoted as \((f \circ g)(x)\), which means you will plug \( g(x) \) into \( f(x) \).To understand this better, let's break it down:

• Take the output of \( g(x) \) and use it as the input for \( f(x) \).
• If \( g(x) = 2x - 1 \) and \( f(x) = x^2 + 3x \), then \((f \circ g)(x) = f(g(x)) = f(2x - 1)\).
• Replace every occurrence of \( x \) in \( f(x) \) with \( 2x - 1 \).

Composition allows you to create new, more complex functions from simpler ones, providing a powerful tool for function transformation and analysis.

Polynomial Functions

Polynomial functions are those functions which are expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants, and \(n\) is a non-negative integer. These functions can take on various shapes based on their degree, which is the highest exponent of \(x\) present.Let's consider an example:

• For the function \(f(x) = x^2 + 3x\), it's a polynomial of degree 2 because the highest power of \(x\) is 2.
• The polynomial is quadratic, which typically forms a parabola when graphed on a Cartesian plane.

Understanding polynomials is essential as they form the basis for many algebraic structures and are fundamental in calculus and real-world applications.

Function Evaluation

Function evaluation refers to the process of finding the output of a function for a given input. This involves substituting the input value for the variable in the function's equation and performing the necessary arithmetic operations.Let's break down the process:

• Take the expression for the function, like \(f(x) = x^2 + x + 1\).
• If you need to evaluate at \(x = 2\), substitute \(2\) everywhere you see \(x\).
• Perform the calculations: \(f(2) = 2^2 + 2 + 1 = 4 + 2 + 1 = 7\).

By evaluating, you find the function's specific value for the chosen input. This is fundamental in understanding function behavior, such as finding intersections, minimums, or maximums on their graphs.