Problem 12 - 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. $$(f+g)(-5)$$

Step-by-Step Solution

Answer

\((f+g)(-5) = -1\).

Step 1: Define the functions

Given the functions, we have:\(f(x) = x^2 + 3x\)\(g(x) = 2x - 1\).

Step 2: Write the expression for \((f+g)(x)\)

The function \((f+g)(x)\) is defined as the sum of \(f(x)\) and \(g(x)\):\((f+g)(x) = f(x) + g(x)\)\((f+g)(x) = (x^2 + 3x) + (2x - 1)\).

Step 3: Simplify \((f+g)(x)\)

To simplify, combine all like terms from \(f(x)\) and \(g(x)\):\((f+g)(x) = x^2 + 3x + 2x - 1\)Combine terms: \(x^2 + 5x - 1\).

Step 4: Substitute \(x=-5\) into \((f+g)(x)\)

Now substitute \(x = -5\) into the simplified function \((f+g)(x) = x^2 + 5x - 1\):\((f+g)(-5) = (-5)^2 + 5(-5) - 1\).

Step 5: Calculate \((f+g)(-5)\)

Perform the calculations:\((-5)^2 = 25\)\(5(-5) = -25\)Put it all together: \((f+g)(-5) = 25 - 25 - 1\)\((f+g)(-5) = -1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a key component in understanding mathematical equations and problems. They consist of numbers, variables, and operations combined together. In the exercise we explored, two algebraic expressions were involved:

• \(f(x) = x^2 + 3x\)
• \(g(x) = 2x - 1\)

These expressions represent the functions \(f\) and \(g\) respectively. A critical step when working with algebraic expressions is combining them through function operations. You add, subtract, multiply, or divide these expressions to explore different outcomes. Remember: make sure to always group and manage similar terms to simplify expressions effectively.

Function Operations

Function operations are processes that allow us to manipulate functions to get new ones. Through addition, subtraction, multiplication, or division, functions can be combined to provide insights into complex problems. In our task:

• We were asked to perform the operation \((f+g)(x)\).
• This meant adding the function expressions \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\).

The goal was to simplify this expression. Adding like terms is crucial here:

• Combine \(3x\) and \(2x\) to get \(5x\).
• The result \((f+g)(x) = x^2 + 5x - 1\) simplifies the polynomial.

Function operations are foundational because they teach us the interconnectedness of algebraic expressions, extending our problem-solving toolbox.

Polynomial Functions

Polynomial functions represent expressions formed by combining variables raised to different powers and constants. Our example showcases a classic polynomial function:

• \(f(x) = x^2 + 5x - 1\)

This function has three terms: \(x^2\), \(5x\), and \(-1\). Here’s what makes polynomial functions noteworthy:

• The degree is determined by the highest power of \(x\), which is 2 in this case.
• They appear neat due to systematically organizing terms from highest to lowest power.

We specifically explored substituting negative values, by plugging \(x = -5\) into the polynomial. Calculating this involved:

• Square the \(-5\).
• Multiply and simplify terms.
• The final value was \(-1\), demonstrating real-world applications of such math concepts.