Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(f(x)=5-2 x^{2}\)

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, \infty)\).

Step 1: Determine Type of Function

The given function is a quadratic function since it is of the form \(f(x) = ax^2 + bx + c\). Quadratic functions are defined for all real numbers.

Step 2: Identify Potential Restrictions

Quadratic functions do not have any restrictions related to undefined points such as division by zero or square roots. Therefore, the domain is not restricted.

Step 3: State the Domain in Interval Notation

Since a quadratic function is defined for all real numbers, the domain in interval notation is \((-\infty, \infty)\).

Key Concepts

Interval Notation

Interval notation is a powerful tool used to represent the set of all possible values of a function's input, called the domain. It provides a compact way to express ranges, making it easier to understand the extent to which a function is applicable. In this notation, the domain of a function is expressed using brackets and parentheses to denote inclusive or exclusive properties. For example:

• \([a, b]\) means the interval includes both endpoints \(a\) and \(b\).
• \((a, b)\) means the interval includes neither endpoint.

To denote infinite ranges, we use the symbols \(-\infty\) and \(\infty\). Always remember that infinity isn’t a number, so it is always represented with parentheses, like \((-\infty, \infty)\), signaling that every real number is included. When working with quadratic functions, interval notation becomes quite useful because these functions often encompass all real numbers.

Quadratic Functions

Quadratic functions are fundamental equations in algebra with the general form \(f(x) = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) squaring makes the function quadratic. These functions describe a parabola when graphed, with crucial characteristics worth noting:

• They can open upwards or downwards depending on the value of \(a\).
• The vertex is the highest or lowest point on the parabola depending on its direction.
• The axis of symmetry divides the parabola into two mirror images.

A remarkable property of quadratic functions is that they are defined for all real numbers, meaning they have no inherent restrictions. This is because there are no divisions by zero or complex square roots involved, so they extend in both directions infinitely. This is why we can confidently state that their domain is \((-\infty, \infty)\).

Function Domains

Understanding the domain of a function is vital in math because it defines what inputs are valid. The domain ensures that any function remains functional without being undefined or problematic in real-world applications. For any given function, whether linear, quadratic, or otherwise, the first task is to identify any restrictions where the function may not remain defined:

• Check for divisions by zero, as these are undefined in mathematics.
• Be cautious of square roots that could result in non-real numbers if negative under the radicand.
• Ensure logarithmic functions don’t operate on non-positive numbers.

For quadratic functions, such as \(f(x) = 5 - 2x^2\), the domain is unrestricted, meaning all real numbers are suitable inputs. The broad applicability of quadratic functions is why their domain is given as \((-\infty, \infty)\). Recognizing these principles helps in comprehending various algebraic functions, representing them accurately, and applying them correctly in problem-solving.