Problem 48 - Calculus: Early Transcendentals

Question

Let \( f(x) = 1/x \) and \( g(x) = 1/x^2 \). (a) Find \( (f \circ g)(x) \). (b) Is \( f \circ g \) continuous everywhere? Explain.

Step-by-Step Solution

Answer

(a) \( (f \circ g)(x) = x^2 \). (b) Yes, it is continuous everywhere.

Step 1: Write the Composition Formula

To find the composition of two functions \( f \) and \( g \), we use the formula for function composition: \((f \circ g)(x) = f(g(x))\).

Step 2: Substitute g(x) into f(x)

Given \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x^2} \), substitute \( g(x) \) into \( f(x) \): \( (f \circ g)(x) = f\left( \frac{1}{x^2} \right) \).

Step 3: Simplify the Expression

Substitute \( \frac{1}{x^2} \) into \( f(x) \). Thus, \( f\left( \frac{1}{x^2} \right) = \frac{1}{\frac{1}{x^2}} \). Simplifying gives \( x^2 \). Therefore, \( (f \circ g)(x) = x^2 \).

Step 4: Check Continuity

To determine if \( f \circ g \) is continuous everywhere, recall that polynomials are continuous everywhere. Since \( f \circ g(x) = x^2 \) is a polynomial, it is continuous everywhere.

Key Concepts

Continuous Functions

Continuous functions are a central topic in calculus and analysis, representing functions that have no jumps, gaps, or breaks in their graphs. Formally, a function is considered continuous at a point if the limit of the function as it approaches the point equals the function's value at that point. This leads to functions that you can "draw" without lifting your pen from the paper.

Key features of continuous functions include:

• No Discontinuities: Continuous functions do not exhibit sudden jumps or breaks.
• Limits Exist: For any continuous function at a point, the limit exists and equals the function's value at that point.
• Easy Composition: The composition of continuous functions is generally continuous, aiding in constructing complex functions from simpler ones.

Understanding continuous functions help in tackling questions about behavior over an interval and can predict how the graph of a function behaves.

Polynomial Functions

Polynomial functions are perhaps the most straightforward continuous functions you will encounter. These functions consist of terms that are combinations of powers of the variable, like in the form: \( ax^n + bx^{n-1} + \, ... \, + c \). Here are some features of polynomial functions:

• Continuous Everywhere: Polynomial functions are continuous across the entire number line due to the algebraic nature of their terms.
• Smooth Graphs: The graph of a polynomial function is a smooth, continuous curve without jumps or sharp angles.
• Well-defined Derivatives: They can be differentiated easily to understand rates of change within their expression.

In our exercise, the function \( (f \circ g)(x) = x^2 \), resulting from the composition, is polynomial. Therefore, it's continuous everywhere as polynomial functions do not break at any point.

Rational Functions

Rational functions are quotients of two polynomial functions, expressed as \( R(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. They can exhibit certain characteristics different from other types of functions due to their fractional nature.

Essential aspects of rational functions include:

• Potential Discontinuity: Rational functions may not be continuous everywhere. They can have discontinuities where the denominator equals zero, resulting in vertical asymptotes.
• Simplification to Polynomials: Under some compositions or constraints, a rational function can reduce to a simpler polynomial, which is then continuous, as seen in the exercise with \( (f \circ g)(x) = x^2 \).
• Domain Restrictions: The domain of rational functions is usually restricted to avoid division by zero. This can complicate their continuity.

Rational functions challenge us to consider the behavior of functions near points of discontinuity, especially important in calculus when evaluating limits and integrals.