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 contimuous 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.

Step 1: Identify the point of discontinuity

The function \( f(x) = \frac{x^2 - 49}{x - 7} \) is not defined at \( x = 7 \) because the denominator becomes zero. At \( x = 7 \), \( f(x) \) leads to division by zero, thus causing discontinuity.

Step 2: Factor the numerator

To work towards continuity, first factor the numerator using the difference of squares: \( x^2 - 49 = (x+7)(x-7) \). So, \( f(x) = \frac{(x+7)(x-7)}{x-7} \).

Step 3: Simplify the function

Cancel the \( x-7 \) term in the numerator and denominator: \( f(x) = x + 7 \), for \( x eq 7 \). This simplification is valid for all \( x eq 7 \).

Step 4: Define the function at the discontinuity

Substitute \( x = 7 \) into the simplified function to find the value that makes \( f(x) \) continuous at \( x = 7 \). We have \( f(7) = 7 + 7 = 14 \). Thus, to make the function continuous at \( x=7 \), define \( f(7) = 14 \).

Key Concepts

Discontinuity

Discontinuity in calculus occurs when a function is not continuous at a certain point. In simpler terms, you can think of it as a "gap" or "jump" in a graph where the function does not smoothly connect.

In the given exercise, the function \( f(x) = \frac{x^2 - 49}{x - 7} \) faces discontinuity at \( x = 7 \) because the denominator becomes zero, which means the function is not defined at that point. Division by zero leads to undefined values in mathematics, creating a break in the function's graph.

• To identify discontinuity, look for points where the denominator is zero.
• At the discontinuity point, the function's limit exists but the function itself is not defined.

To resolve this, we redefine the function at the point by finding a value that makes it continuous, ensuring the graph becomes a smooth, unbroken line across the entire domain.

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that breaks down a complex polynomial expression into simpler, multiplied factors. This process is useful for simplifying expressions and resolving discontinuities like in the given problem.

In the formula \( f(x) = \frac{x^2 - 49}{x - 7} \), the polynomial in the numerator, \( x^2 - 49 \), can be factored.

• Recognize common polynomial identities, such as the difference of squares, to factor effectively.
• Look for pairs of terms that can be grouped and factored out.

Factoring \( x^2 - 49 \) using the difference of squares yields \((x+7)(x-7)\).

This transformation simplifies \( f(x) \) and helps in canceling out terms, paving the way for continuity at the initially problematic point.

Difference of Squares

The difference of squares is a special factoring technique useful in algebraic manipulations. It is based on the identity \( a^2 - b^2 = (a+b)(a-b) \), where two squares are subtracted to result in a product of a sum and a difference.

In the problem \( f(x) = \frac{x^2 - 49}{x - 7} \), the expression \( x^2 - 49 \) is a classic example, as it conforms to the difference of squares identity where \( a = x \) and \( b = 7 \).

• This factoring technique helps simplify expressions by removing squares and direct comparisons.
• Enables cancellation of like terms in fractions simplifying solutions.

By recognizing \( x^2 - 49 \) as a difference of squares, you factor it into \((x+7)(x-7)\), thus canceling \(x-7\) in the numerator and denominator of \( f(x) \), which simplifies the discontinuous point enabling assignment of a specific value to maintain continuity.