Problem 23 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$g(x)=x^{2}+2 x+3$$ a. \(g(-1)\) b. \(g(x+5) \quad\) c. \(g(-x)\)

Step-by-Step Solution

Answer

The values are a. \(g(-1)=2\), b. \(g(x+5) = x^{2}+12x+38\), c. \(g(-x) = x^{2}-2x+3\).

Step 1: Evaluate g(-1)

Substitute \(x = -1\) into the equation \(g(x)=x^{2}+2x+3\). This gives \(g(-1)=(-1)^{2}+2*(-1)+3=1-2+3=2\).

Step 2: Evaluate g(x+5)

Substitute \(x = x+5\) into the equation \(g(x)=x^{2}+2x+3\). This gives \(g(x+5)=(x+5)^{2}+2*(x+5)+3=(x^{2}+10x+25)+2x+10+3=x^{2}+12x+38\).

Step 3: Evaluate g(-x)

Substitute \(x = -x\) into the equation \(g(x)=x^{2}+2x+3\). This gives \(g(-x)=(-x)^{2}+2*(-x)+3=x^{2}-2x+3\).

Key Concepts

Independent Variable

In mathematics, especially when dealing with functions, an independent variable is the input of the function. For the function described as \( g(x) = x^2 + 2x + 3 \), the independent variable is \(x\). It simply acts as a placeholder or a symbol representing the input value. When evaluating a function at specific points, you substitute these points into the equation wherever the independent variable appears.

• For example, in part (a), substituting \(x = -1\) leads to \(g(-1) = 1 - 2 + 3 = 2\).
• In part (b), substituting \(x = x + 5\) generalizes the input, making the function \(g(x+5) = x^2 + 12x + 38\).
• Similarly, evaluating \(g(-x)\) in part (c) changes the input to its negative counterpart, resulting in \(x^2 - 2x + 3\).

Understanding the role of the independent variable helps in manipulating and evaluating functions easily.

Function Simplification

Simplifying a function involves reducing the expression to its most basic form without changing its value or outcome. The goal is to make the function more manageable and easier to work with while maintaining its integrity.
To simplify correctly, you must follow basic algebraic rules like combining like terms and applying distributive properties. Take, for instance, the simplification of \(g(x+5)\):
- First, expand \((x+5)^2\) which results in \(x^2 + 10x + 25\).
- Next, distribute the input \(2(x + 5)\) to get \(2x + 10\).
- Combine these with the constant \(+3\) to finally yield \(x^2 + 12x + 38\).
By understanding each step and the algebra involved, simplifying functions can become a smoother, more logical process. These fundamentals are often applied when processing more complex mathematical models and systems.

Polynomial Functions

Polynomial functions are a cornerstone of algebra, relating to equations that are expressed as a sum of terms including the independent variable raised to various powers. The function \(g(x) = x^2 + 2x + 3\) is a quadratic polynomial, meaning it is a second-degree polynomial function.

• The degree of this polynomial is 2, as indicated by the highest power of \(x\).
• All polynomial functions are continuous and smooth, meaning they have no breaks or sharp corners on their graphs.
• They can be classified by their degree: linear, quadratic, cubic, quartic, and so forth.

Each type of polynomial has distinct characteristics in terms of its graph shape and intercepts. Quadratic polynomials, for example, typically form parabolas. Understanding how to manipulate and evaluate polynomial functions is crucial for solving complex algebraic problems, making them an essential part of the pre-calculus curriculum.