Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{1}{4+x^{2}}\)

Step-by-Step Solution

Answer

The function \(f(x)=\frac{1}{4+x^{2}}\) is continuous for all real numbers.

Step 1: Identify Possible Values for Denominator Equals Zero

Solve the equation \(4+x^{2}=0\) to find any values of \(x\) that could possibly make the function undefined.

Step 2: Solve the Equation

Solving for \(x\), gives \(x = \pm \sqrt {-4}\), which are complex numbers. This means for any real number \(x\), the denominator never equals zero.

Step 3: Determine the Continuity

Since no real values of \(x\) can make the denominator zero, the function is defined for all real numbers. And we know that a function f is continuous everywhere in its domain, thus the function \(f(x)=\frac{1}{4+x^{2}}\) is continuous for all real numbers.

Key Concepts

Real Numbers

The world of real numbers can be a bit vast, but it fundamentally refers to all the numbers that exist on the number line. Real numbers encompass both rational numbers (like 1/2 or 2) and irrational numbers (such as \( \sqrt{2} \) or \( \pi \)).

• Rational numbers are all the numbers that can be expressed as a fraction.
• Irrational numbers are those that cannot be perfectly expressed as fractions.
• Both types combined form the set of real numbers.

Real numbers are crucial in mathematics because we encounter them everywhere in real life, whether calculating the height of a building or measuring the speed of a car. They help us understand and plot functions that describe real-world situations. In the exercise, we see that the function is analyzed within the context of real numbers, as we examine where the function might be undefined or continuous across this number system.

Complex Numbers

Complex numbers expand the idea of traditional number systems by including solutions to equations that do not have real solutions. A complex number is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

• They are useful for solving equations like \( x^2 + 1 = 0 \), which has no solution among real numbers but has solutions \( x = i \) and \( x = -i \).
• Complex numbers include real numbers as a subset, where the imaginary part is zero.

In the exercise, when solving the equation \( 4 + x^2 = 0 \), the solutions involve complex numbers \( x = \pm \sqrt{-4} \), which are not part of the real number set. Hence, for any real value of \( x \), the function's denominator does not become zero, aligning with the function's continuity over all real numbers.

Continuous Functions

A continuous function is one where small changes in the input (in this case, \( x \)) result in small changes in the output (\( f(x) \)). Simply put, you can draw the graph of a continuous function without lifting your pencil. In analyzing a function's continuity, especially over the real numbers, it's essential to check that the function is well-defined and behaves predictably.

• A function is continuous if it does not have any breaks, holes, or jumps on its graph.
• In mathematical terms, if a function is defined, finite, and returns a value for every input in its domain, it is continuous over that domain.

The exercise focuses on ensuring that the function is continuous for all real numbers. Since \( f(x) = \frac{1}{4 + x^2} \) does not have any real values for which the denominator becomes zero, it remains continuous across the entire set of real numbers. This analysis helps confirm that the function's behavior is stable and predictable as \( x \) varies within its domain.