Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(24-35\), at what points, if any, are the functions discontinuous? $$ f(x)=\frac{3 x+7}{(x-30)(x-\pi)} $$

Step-by-Step Solution

Answer

The function is discontinuous at \(x=30\) and \(x=\pi\).

Step 1: Identify the form of the function

The given function is a rational function, expressed as \( f(x) = \frac{3x+7}{(x-30)(x-\pi)} \). Notice that rational functions are generally continuous except where the denominator is zero.

Step 2: Determine points where the denominator is zero

Set the denominator \((x-30)(x-\pi)\) equal to zero to find the points of discontinuity. Solve \((x-30)(x-\pi) = 0\). This equation is satisfied when either \(x-30=0\) or \(x-\pi=0\).

Step 3: Solve for points of discontinuity

Solve the equations \(x - 30 = 0\) to get \(x = 30\), and \(x - \pi = 0\) to get \(x = \pi\). These are the points at which the denominator becomes zero, thus making the function discontinuous.

Key Concepts

Rational functions

A rational function is a type of function that is formed by dividing two polynomial functions. Essentially, it’s expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \), the denominator, is not zero. This means:

• The numerator \( P(x) \) could be any polynomial like a linear \( ax+b \), quadratic \( ax^2+bx+c \), or higher-order polynomial.
• The denominator \( Q(x) \) must also be a polynomial expression, but it can never be zero because division by zero is undefined.

Rational functions are often continuous over their domain except at the points where the denominator equals zero. This property makes finding such functions' discontinuities straightforward. Simply identify when the denominator hits zero, and you have your discontinuity points. These points are crucial in calculus because they represent limits or boundaries where the function behaves differently or doesn’t exist.

Discontinuity points

Discontinuity points in a function are those particular values of \( x \) where a function does not behave "nicely". For rational functions, a discontinuity occurs primarily due to division by zero in the denominator. To find these points, you:

• Set the denominator equal to zero.
• Solve for \( x \) to obtain possible points of discontinuity.

In our exercise with the function \( f(x) = \frac{3x+7}{(x-30)(x-\pi)} \), the discontinuities arise at points \( x = 30 \) and \( x = \pi \). These are the values making the denominator zero, thus making the function undefined and discontinuous at these points. However, outside of these points, the function can continue seamlessly, remaining continuous as rational functions typically are over their valid domains.

Denominator analysis

Analyzing the denominator of a rational function is key to understanding its behavior, specifically in identifying discontinuity points. Here, the denominator is \((x-30)(x-\pi)\). The task is straightforward:

• Determine the values of \( x \) that make each factor zero.
• Solve \( x-30=0 \) which gives \( x=30 \).
• Solve \( x-\pi=0 \) which gives \( x=\pi \).

These solutions correspond to potential discontinuities, as these are the values at which the function’s denominator becomes zero, leading the function itself to become undefined. Understanding the behavior of the function at these critical points is essential in calculus. It helps in drawing accurate graph interpretations, pointing out where functions may transition between defined and undefined behavior, or where limits may or may not exist.