Problem 74 - Elementary and Intermediate Algebra

Question

Let \(f(x)=x^{2}-1\) and \(g(x)=x^{2}-4 .\) Find each function and give its domain. $$ f+g $$

Step-by-Step Solution

Answer

\((f+g)(x) = 2x^2 - 5\) with domain \(\mathbb{R}\).

Step 1: Identify Each Function

The functions given are \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\). These are both quadratic functions.

Step 2: Adding the Functions

To find \(f+g\), we need to sum the functions: \((f+g)(x) = f(x) + g(x)\). Therefore, we have:\[(f+g)(x) = (x^2 - 1) + (x^2 - 4)\]\[= x^2 - 1 + x^2 - 4\]\[= 2x^2 - 5\].

Step 3: Determine the Domain of the Sum

Since \(f(x)\) and \(g(x)\) are both polynomials, the domain of each function is all real numbers \(\mathbb{R}\). When adding two polynomials, the domain remains the same. Therefore, the domain of \((f+g)(x)\) is also all real numbers \(\mathbb{R}\).

Key Concepts

Function Arithmetic

Function arithmetic involves performing basic operations like addition, subtraction, multiplication, and division on functions. This concept allows us to combine functions to create new ones. Consider two functions, say \( f(x) \) and \( g(x) \). If you want to add these two, you simply add their corresponding outputs. In formula terms, the sum \((f+g)(x)\) is calculated as \(f(x) + g(x)\).

Here's a step-by-step breakdown of how it works:

• Identify each function and their outputs.
• Add the outputs together to create a new function.
• Simplify the resulting expression if possible.

When given two quadratic functions, as in the exercise with \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\), you perform these steps by adding their similar terms. That's how we derived the new function \((f+g)(x) = 2x^2 - 5\).

This new function is itself another quadratic function, showcasing that polynomial functions are closed under addition—meaning that adding polynomials always results in a polynomial.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable that the function can accept. For most polynomial functions, including linear, quadratic, cubic, and so on, this set of values is typically all real numbers \(\mathbb{R}\). This is because polynomial functions are defined for every real number input, meaning there's no value that will result in an undefined situation.

For the exercise's functions \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\), since they are both quadratic polynomials, their domains are all real numbers \(\mathbb{R}\).

When these functions are added, the resulting function \((f+g)(x) = 2x^2 - 5\) also inherits this domain. Therefore,

• The domain of \(f(x)\) is \(\mathbb{R}\).
• The domain of \(g(x)\) is \(\mathbb{R}\).
• Thus, the domain of \((f+g)(x)\) is also \(\mathbb{R}\).

By understanding domains, you ensure that the function is always operating within permissible values, helping you avoid errors and undefined scenarios.

Polynomial Functions

Polynomial functions are fundamental mathematical expressions involving variables raised to whole number powers. These functions are known for producing smooth, continuous curves without breaks or cusps on a graph. A general polynomial can be written as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n\), \(a_{n-1}\), etc., are coefficients and \(n\) is a non-negative integer.

Characteristics of polynomial functions include:

• Smoothness: The graph of every polynomial function is smooth and continuous.
• Closure under operations: Adding, subtracting, or multiplying polynomials always results in a polynomial.
• Domain: All real numbers \(\mathbb{R}\) can be input into polynomial functions.

In our exercise, both \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\) are quadratic polynomials. When added, they form the new polynomial \((f+g)(x) = 2x^2 - 5\). This equation is another quadratic function and exemplifies a simple yet powerful application of polynomial arithmetic. Quadratic polynomials like these often appear as parabolic shapes on graphs, opening upwards or downwards depending on the sign of the coefficient in front of \(x^2\).