Problem 27 - Calculus: Early Transcendentals

Question

Continuity Determine the interval(s) on which the following functions are continuous. $$f(x)=\frac{x^{5}+6 x+17}{x^{2}-9}$$

Step-by-Step Solution

Answer

Answer: The function is continuous on the intervals $$(-\infty, -3), (-3, 3)$$ and $$(3, \infty)$$

Step 1: Identify the function type

The given function is a rational function of the form: $$f(x)=\frac{x^{5}+6 x+17}{x^{2}-9}$$

Step 2: Find the domain of the function

To find the domain of the function, we need to find the values of x for which the denominator is nonzero. Thus, we need to find the points where $$x^2 - 9 \neq 0$$ Rearrange and we get: $$x^2 \neq 9$$ Solving for x gives us two solutions, $$x \neq \pm3$$

Step 3: Identify the intervals of continuity

Now that we know the values of x where the function is not continuous, we can find the intervals of continuity. As the function is continuous everywhere except x=-3 and x=3, we can define the intervals as: 1. $$(-\infty, -3)$$ 2. $$(-3, 3)$$ 3. $$(3, \infty)$$ Thus, the function is continuous on the intervals $$(-\infty, -3), (-3, 3)$$ and $$(3, \infty)$$.

Key Concepts

Rational Functions

Rational functions are mathematical expressions defined by the ratio of two polynomials. The general form of a rational function can be expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) \eq 0\). The behavior and properties of a rational function are largely determined by its numerator and, most importantly, its denominator.

Rational functions can exhibit a variety of features including asymptotes, intercepts, and intervals of increase and decrease. Importantly for our discussion, the points where the denominator equals zero are not part of the function's domain and for real-valued functions, these points typically lead to vertical asymptotes where the function values approach infinity.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable, typically denoted as \(x\), for which the function is defined. This concept is vital in understanding any function's behavior.

For rational functions, the domain specifically excludes any values that would make the denominator zero, since division by zero is undefined. By determining where the denominator is nonzero, you establish the domain of the function. This analysis prevents mathematical impossibilities and allows for an accurate description of the function's behavior over its entire range.

Finding the Domain

To find the domain for a given function, you must look for values that would either make the denominator zero or lead to a negative number under an even root, as in the case of square roots, amongst other considerations. For instance, our original function \( f(x) = \frac{x^{5}+6 x+17}{x^{2}-9} \) excludes \( x = \pm 3 \) from its domain because these values zero out the denominator.

Intervals of Continuity

Intervals of continuity refer to the sections of the domain of a function where it is defined and does not have any breaks, jumps, or holes. A function is continuous on an interval if, for every point within that interval, the function is defined, and its graph can be drawn without lifting the pen from the paper.

For a rational function, the intervals of continuity can be found by removing any points that cause discontinuity—primarily the zeroes of the denominator. Once these points are identified and excluded, the remaining intervals of the function's domain represent where the function is continuous.

Examining Continuity

Continuity of a function is significant as it ensures the predictability of the function's behavior within those intervals. In practical terms, for our function \(f(x)\), continuity is confirmed on the intervals \( (-\infty, -3) \), \( (-3, 3) \), and \( (3, \infty) \) because these are the regions where the function's denominator does not become zero, avoiding any undefined behavior.