Problem 2 - A Graphical Approach to College Algebra

Question

Provide a short answer to each question. Do not use a calculator. What is the domain of \(f(x)=\frac{1}{x^{2}} ?\) What is its range?

Step-by-Step Solution

Answer

Domain: \((-\infty, 0) \cup (0, \infty)\), Range: \((0, \infty)\).

Step 1: Identify the Domain of the Function

The domain of a function consists of all the possible input values (x-values) for which the function is defined. For the given function \(f(x) = \frac{1}{x^2}\), the denominator cannot be zero, because division by zero is undefined. Therefore, set the denominator equal to zero and solve for \(x\): \(x^2 = 0\). This gives \(x = 0\). Hence, the function is undefined at \(x = 0\). The domain consists of all real numbers except zero. In interval notation, this is \((-\infty, 0) \cup (0, \infty)\).

Step 2: Determine the Range of the Function

The range of a function consists of all possible output values (y-values) of the function. For \(f(x) = \frac{1}{x^2}\), the value of \(x^2\) is always positive for any nonzero real \(x\). Therefore, \(\frac{1}{x^2}\) will always be positive. As \(x\) approaches zero from either direction, \(f(x)\) tends to infinity, and as \(|x|\) becomes very large, \(f(x)\) approaches zero. Thus, the range of the function is all positive real numbers, expressed in interval notation as \((0, \infty)\).

Key Concepts

Function Notation

Function notation is a way of representing functions in mathematics. It expresses the relation between a set of inputs and a set of permissible outputs. Here's how it breaks down:

• In function notation, a function is typically written as \(f(x)\), where \(f\) names the function and \(x\) represents the variable or input value.
• The expression after the equal sign shows the operation or expression that will be performed on the input \(x\). For example, in the function \(f(x) = \frac{1}{x^2}\), \(\frac{1}{x^2}\) describes the operation.

Function notation provides a clear and concise way to show operations that are applied to inputs to produce outputs. It's a handy tool not only for algebra but also for calculus and other advanced branches of mathematics.

Interval Notation

Interval notation is a method of describing sets of numbers, typically to represent the domain or range of a function. It uses intervals to show where these sets begin and end. Here's a guide on how it works:

• Parentheses \(()\) are used to indicate that endpoints are not included in the interval. For instance, the interval \((0, \infty)\) includes all positive numbers but not zero.
• Brackets \([]\) signify that endpoints are included; for example, \([0, 1]\) includes both 0 and 1 along with all numbers in between.

In the provided function \(f(x) = \frac{1}{x^2}\), the domain is all real numbers except zero, written as \(( -\infty, 0 ) \cup (0, \infty)\). This means every number is included except zero. Understanding interval notation helps in accurately defining the scope of functions.

Real Numbers

Real numbers consist of all rational and irrational numbers, embodying every point on the number line. They are essential in defining domains and ranges in functions. Let's explore what they are:

• Rational numbers are fractions or decimals that terminate or repeat, like \(\frac{1}{2}\) or 0.333\