Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=\frac{x^{2}-4}{x^{2}-4} .\) Solve each equation or inequality. (a) \(f(x)=0\) (b) \(f(x)<0\) (c) \(f(x)>0\) (d) \(f(x)=1\)

Step-by-Step Solution

Answer

(a) No solution; (b) No solution; (c) \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \); (d) \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).

Step 1: Understanding the function

The function given is \( f(x) = \frac{x^2 - 4}{x^2 - 4} \). For any \( x \) not equal to 2 or -2, the function simplifies to 1, because the numerator and denominator are equal. However, for \( x = 2 \) or \( x = -2 \), the function is undefined since you cannot divide by zero.

Step 2: Solving \( f(x) = 0 \)

The equation \( f(x) = 0 \) implies \( \frac{x^2 - 4}{x^2 - 4} = 0 \). However, for the fraction \( \frac{x^2-4}{x^2-4} \) to equal zero, the numerator would need to be zero while the denominator is not, which is impossible here. Thus, there is no solution for \( f(x) = 0 \).

Step 3: Solving \( f(x) < 0 \)

The inequality \( f(x) < 0 \) implies \( \frac{x^2 - 4}{x^2 - 4} < 0 \). This is also impossible because when the function is defined (for \( x eq 2 \text{ or } -2 \)), it equals 1, not a negative number. Hence, there are no solutions for \( f(x) < 0 \).

Step 4: Solving \( f(x) > 0 \)

The inequality \( f(x) > 0 \) implies \( \frac{x^2 - 4}{x^2 - 4} > 0 \). This is true for any \( x \) except \( x = 2 \) or \( x = -2 \), where the function is undefined. Therefore, the solution is \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).

Step 5: Solving \( f(x) = 1 \)

The equation \( f(x) = 1 \) implies \( \frac{x^2 - 4}{x^2 - 4} = 1 \). This is true for \( x eq 2 \text{ or } -2 \), as the function simplifies to 1 where it is defined. Thus, the solution is \( x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).

Key Concepts

Undefined Function

In mathematics, a function becomes undefined where divisions by zero occur. In the case of our function \( f(x) = \frac{x^2 - 4}{x^2 - 4} \), this happens when \( x^2 - 4 = 0 \). This expression can be factored as \( (x - 2)(x + 2) = 0 \). Hence, the roots are \( x = 2 \) and \( x = -2 \), making \( x = 2 \) and \( x = -2 \) points where the function is undefined. Since division by zero is not possible, these values make the function not interpretable in real numbers.

**Key Takeaways:**

• A function is undefined at any points where the denominator equals zero.
• To find undefined points, solve the equation of the denominator set to zero.
• Undefined points can cause discontinuities in graphs and solutions.

Rational Expressions

Rational expressions are fractions with polynomials in both the numerator and the denominator. In our function, \( f(x) = \frac{x^2 - 4}{x^2 - 4} \), both the numerator and the denominator are the same polynomial. This similarity means the expression reduces to 1 where it is defined. An important concept to note is that simplifying rational expressions involves canceling out common factors, but also requires keen attention to where the expression might be undefined due to zero denominators.

**Key Pointers on Rational Expressions:**

• To simplify, factor both the numerator and the denominator, and cancel out the common factors.
• Simplification must respect points where the expression is undefined.
• Simplified expressions can help solve equations and inequalities effectively.

Inequalities in Mathematics

In solving inequalities related to functions like \( f(x) = \frac{x^2 - 4}{x^2 - 4} \), careful analysis is crucial. For example, for \( f(x) < 0 \), or for \( f(x) > 0 \), it is key that we only evaluate these inequalities where the function is defined. Since the function reduces to 1 except where \( x = 2 \) or \( x = -2 \), you can determine when \( f(x) \) is positive or undefined but cannot find when it is negative.

**Understanding Inequalities in Mathematics:**

• Evaluate inequalities only within the domain of the defined function.
• Recognize that rational expressions like these will require finding and excluding undefined points from solutions.
• Solved inequalities often express intervals reflecting where the solution is valid.