Problem 74 - 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

The function is \((f+g)(x) = 2x^2 - 5\) and its domain is \(\mathbb{R}\).

Step 1: Understand the Given Functions

We have two functions: \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \). Both are polynomial functions of degree 2, which are quadratic and defined for all real numbers. Their domains are both \( \mathbb{R} \).

Step 2: Find the Sum of the Functions

We find \( (f + g)(x) \) by adding the two given functions: \[ (f + g)(x) = f(x) + g(x) = (x^2 - 1) + (x^2 - 4) = 2x^2 - 5. \]

Step 3: Determine the Domain of the Sum Function

Since both \( f(x) \) and \( g(x) \) are polynomials, their sum, \( 2x^2 - 5 \), is also a polynomial. The domain of a polynomial function is all real numbers. Thus, the domain of \( f + g \) is \( \mathbb{R} \).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental part of algebra. They take the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions are known for their characteristic parabolic curves in graphs. Quadratics are also important because they include operations like squaring, which is essential in many fields of mathematics.

• The leading term, \( ax^2 \), determines the direction of the parabola. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.


• The vertex of the parabola provides the maximum or minimum point, depending on the parabola's orientation.


• The roots, or values of \( x \) where the parabola crosses the x-axis, can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Example: In our functions \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \), both have the highest degree of 2, making them quadratic. Their graphs are parabolas centered along the y-axis.

Domain of a Function

The domain of a function includes all possible input values (\( x \) values) that the function can accept. Understanding the domain is crucial as it defines where the function is valid.

• For polynomial functions, including quadratic functions, the domain is typically all real numbers \( \mathbb{R} \). This means that any real number can be plugged into the function, and the function will output a valid result.


• Other types of functions, like square roots or rational expressions, may have restricted domains due to division by zero or negative square roots.

In the exercise, both \( f(x) \) and \( g(x) \) are polynomial functions, so their domains are all real numbers \( \mathbb{R} \). This rule applies to the sum of these functions, \( 2x^2 - 5 \), which is also a polynomial.

Function Operations

Performing operations on functions is an essential skill. It involves adding, subtracting, multiplying, or dividing functions to create new functions.

Addition of Functions:

When we add two functions, we simply add their outputs for every input value. For example, if \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \), then \( (f+g)(x) = f(x) + g(x) \). Here, each function value is summed individually, resulting in \( 2x^2 - 5 \).

• The sum function combines terms that have the same degree.


• Common terms are combined by adding their coefficients.

Domains in Function Operations:

The domain of the resulting function after an operation usually includes the intersection of the domains of all involved functions. Since both functions in the given exercise have domains of all real numbers, their sum \( 2x^2 - 5 \) also has a domain of all real numbers \( \mathbb{R} \).
Understanding these operations and their effects on function domains is critical for working accurately with functions in algebra.