Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 18-23, the given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? (See Example 1.) $$ f(x)=\frac{x^{2}-49}{x-7} $$

Step-by-Step Solution

Answer

Define \( f(7) = 14 \) to make the function continuous at \( x = 7 \).

Step 1: Identify the point of discontinuity

The function \( f(x) = \frac{x^2 - 49}{x - 7} \) is discontinuous at \( x = 7 \) because the denominator becomes zero, making the expression undefined.

Step 2: Simplify the function

Factor the numerator: \( x^2 - 49 \) is a difference of squares. Thus, \( x^2 - 49 = (x-7)(x+7) \). The function becomes \( f(x) = \frac{(x-7)(x+7)}{x-7} \).

Step 3: Cancel common factors

The term \( (x-7) \) appears in both the numerator and denominator, except where \( x = 7 \). Cancel these terms to simplify the function to \( f(x) = x + 7 \) for all \( x eq 7 \).

Step 4: Define the function at \( x = 7 \)

To make \( f(x) \) continuous at \( x = 7 \), define \( f(x) \) such that its limit as \( x \) approaches 7 equals the function value at \( x = 7 \). Calculate the limit: \( \lim_{{x \to 7}} f(x) = 7 + 7 = 14 \). Therefore, define \( f(7) = 14 \).

Key Concepts

Discontinuity

Discontinuity happens when a function cannot be smoothly followed along its entire domain due to some interruption or gap. In the case of the function \( f(x) = \frac{x^2 - 49}{x - 7} \), the problem lies at \( x = 7 \) because the denominator becomes zero. When a denominator of a fraction within a function becomes zero, it leads to an undefined situation, creating a point of discontinuity.

In practical terms, this means that there is a break in the graph of the function at \( x = 7 \), making it impossible to draw the function without lifting the pencil from the paper at that point. The goal, in such cases, is to redefine a function carefully to ensure it becomes continuous, filling in any gaps or jumps at the troublesome point.

Limits

Limits are a fundamental concept in calculus and they help us understand the behavior of a function as it approaches a particular point. Using limits, we can determine the value that a function is approaching even if it is not defined there.

In our example with \( f(x) = \frac{x^2 - 49}{x - 7} \), we determined the limit as \( x \) approaches 7. Since after simplifying, \( f(x) = x + 7 \) wherever \( x eq 7 \), the limit as \( x \to 7 \) becomes \( 14 \).

This means, although \( x = 7 \) is not initially in the domain (because of division by zero), the output of the function is always tending towards 14 as \( x \) approaches 7. By defining \( f(7) = 14 \), we ensure the function is continuous at this point.

Factorization

Factorization is the method of breaking down expressions into a product of simpler factors. In the given function, the numerator \( x^2 - 49 \) can be rewritten using the difference of squares method. This polynomial identity states that \( a^2 - b^2 = (a-b)(a+b) \).

Applying this to \( x^2 - 49 \), we get \((x-7)(x+7)\). By factorizing, we simplify the expression so that when it is divided by \( x-7 \) in the denominator, the troublesome \( x-7 \) terms can be canceled out, leaving the much simpler function \( x+7 \) for \( x eq 7 \).

Factorization thus helps to clarify situations where continuity can be re-established by removing apparantly complicating elements in the function's expression.

Functions

Functions are mathematical objects that relate a set of inputs to a set of outputs. Every function \( f(x) \) maps each element from its domain to a unique element in its range. The behavior of functions can vary greatly depending on their defining rules.

In the function \( f(x) = \frac{x^2 - 49}{x - 7} \), initially, there seems to be a problem due to the discontinuity at \( x = 7 \). However, by altering the rules slightly – through simplification – we can redefine this function's behavior at that point to maintain continuity.

The essential nature of functions is to provide a predictable and consistent mapping from input to output, and addressing discontinuities through methods such as defining new points or simplifying expressions is crucial in securing that stability across all inputs.