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

Step-by-Step Solution

Answer

The result is -1.

Step 1: Understand the Problem Statement

We are asked to find \((f+g)(x)\) and then evaluate it at \(x = -5\). "Function addition" involves adding the function values for each function separately and evaluating the result for a given x-value.

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

The expression \((f+g)(x)\) is defined as \(f(x) + g(x)\). For the given functions, it is \((x^2 + 3x) + (2x - 1)\).

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

Now we simplify the expression: \[(f+g)(x) = x^2 + 3x + 2x - 1 = x^2 + 5x - 1\].

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

Substitute \(-5\) into the simplified expression: \((f+g)(-5) = (-5)^2 + 5(-5) - 1\).

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

First, calculate \((-5)^2\) which is 25. Then calculate \(5(-5)\) which is -25. Finally we substitute: \((f+g)(-5) = 25 - 25 - 1 = -1\).

Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the result of another function. Imagine having two functions, such as \(f(x)\) and \(g(x)\). Compose them by substituting \(g(x)\) into \(f(x)\) to find \(f(g(x))\). It’s like nesting operations or calculations. When performing function compositions:

• First, substitute the inside function into the outer function.
• Always pay attention to the order, as \(f(g(x))\) is different from \(g(f(x))\).

For instance, for \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\), calculate \(f(g(x))\): substitute \(2x - 1\) into \(f(x)\). This means everywhere you see \(x\) in \(f(x)\), replace it with \(2x - 1\). Then, calculate the new expression.Function composition is a powerful tool in mathematics, serving in solving complex problems by simplifying multiple functions into a single expression.

Function Addition

Function addition is the process of adding two functions together. This results in a new function that combines the effects of both. For instance, suppose you have two functions \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\).To find \((f+g)(x)\), simply add the expressions of \(f(x)\) and \(g(x)\):

• Combine like terms: \(f(x) + g(x) = (x^2 + 3x) + (2x - 1)\).
• Simplify the expression by adding the coefficients of similar terms: In this case, the result is \(x^2 + 5x - 1\).

Function addition is simply and effectively combining functions to form new expressions. It is straightforward as long as you carefully align and combine like terms. Finally, evaluate the new function at specific values to find distinct outputs. This was demonstrated by computing \((f+g)(-5)\), resulting in \(-1\).

Polynomial Functions

Polynomial functions are expressions made up of terms called monomials added together. Each monomial is a product of a constant and a whole number power of \(x\). For example, a polynomial function like \(f(x) = x^2 + 3x\) consists of two terms: one quadratic term \(x^2\) and one linear term \(3x\).Key characteristics of polynomial functions:

• The degree is determined by the highest power of \(x\) (e.g., \(x^2\) means the polynomial is of degree 2).
• They can have different forms, such as linear (degree 1), quadratic (degree 2), or cubic (degree 3).
• Polynomial functions are smooth and continuous, making them useful in a variety of applications, from simple equations to modeling complex systems.

The function \((f+g)(x) = x^2 + 5x - 1\) combines terms from both \(f(x)\) and \(g(x)\), showcasing polynomial operations in action. Recognizing and working with polynomials is essential in algebra and calculus as they form the basis for many mathematical models.