Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group II. $$I$$ $$(f+g)(x)$$ $$\mathbf{II}$$ $$A.\quad4 x^{2}-20 x+25$$ $$B.\quad x^{2}-2 x+5$$ $$C.\quad 2 x^{2}-5$$ $$D.\quad \frac{x^{2}}{2 x-5}$$ $$E. \quad x^{2}+2 x-5$$ $$F. \quad 2 x^{3}-5 x^{2}$$

Step-by-Step Solution

Answer

(f+g)(x) matches with option E: \(x^2 + 2x - 5\).

Step 1: Understand the Problem

We need to find an expression for \((f+g)(x)\) where \(f(x) = x^2\) and \(g(x) = 2x - 5\). This means adding the functions \(f(x)\) and \(g(x)\) together.

Step 2: Substitute the Functions

The expression \((f+g)(x)\) means we need to substitute \(f(x) = x^2\) and \(g(x) = 2x - 5\). Therefore, \((f+g)(x)\) becomes \((x^2) + (2x - 5)\).

Step 3: Simplify the Expression

Now, combine like terms: \(x^2 + 2x - 5\). This is the simplified form of \((f+g)(x)\).

Step 4: Match with Group II

Compare the expression \(x^2 + 2x - 5\) with the expressions in Group II. This matches with option E: \(x^2 + 2x - 5\).

Key Concepts

Polynomial Functions

Polynomial functions are fundamental in understanding algebra. They are composed of a sum of terms, each consisting of a variable raised to a power and multiplied by a coefficient. In general, a polynomial function can be expressed as follows:

• The basic form: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)
• Each term, like \( a_ix^i \), represents a part of the polynomial function.
• The highest power of the variable is called the degree of the polynomial.

In our exercise, the polynomial function \( f(x) = x^2 \) is a simple quadratic polynomial with a degree of 2 since the highest power is 2. This simplicity helps in learning basic operations like function addition, where such polynomials are combined with other types of functions, like linear functions. Understanding this core structure is essential for progressing in algebra.

Expressions Simplification

Simplifying expressions in mathematics means reducing them to their simplest form, which involves combining like terms and performing any possible arithmetic operations. In the context of the given exercise, simplification took place when we combined the terms from \( f(x) \) and \( g(x) \):

• First, list out all terms: \( x^2, 2x, \text{ and } -5 \).
• No further like terms can be combined since they all have different degrees.
• The result, \( x^2 + 2x - 5 \), is the simplest form of the function addition \((f+g)(x)\).

Simplification is key for solving equations and functions because it presents the expression in a more manageable form, making it easier to work with or compare with other mathematical expressions. The goal is to make mathematics clearer and more intuitive, especially when performing function operations like in our example.

Function Operations

Function operations include addition, subtraction, multiplication, and division of functions. These operations combine two or more functions to produce a new function, allowing for a broader analysis of mathematical models. In our exercise, the operation in focus is function addition:

• The addition of two functions \( (f+g)(x) \) involves adding their outputs for any input \( x \): \( f(x) + g(x) \).
• We take \( f(x) = x^2 \) and \( g(x) = 2x - 5 \), and compute \( (f+g)(x) = x^2 + 2x - 5 \).
• This operation illustrates how functions can be combined to form a new function that inherits characteristics from each function, such as shape and intercepts.

Function operations allow us to explore complex relationships between variables and serve as tools for modeling and solving real-world problems. Mastering these operations is crucial as they lay the groundwork for more advanced studies in mathematics.