Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Sketch the graph of \(f .\) (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing or is decreasing. $$ f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

(a) Sketch shows hyperbola with asymptotes. (b) Domain: \( \mathbb{R} \setminus \{0\} \), Range: \( (0, \infty) \). (c) Increasing on \( (-\infty, 0) \), decreasing on \( (0, \infty) \).

Step 1: Graph the Function

The function given is \( f(x) = \frac{1}{x^2} \). To sketch this graph, recognize that it's a reciprocal quadratic function. It will have a vertical asymptote at \( x = 0 \) because the function is undefined there, and a horizontal asymptote at \( y = 0 \) as \( x \) tends to positive or negative infinity. The graph has symmetry about the y-axis (even function symmetry) and is always above the x-axis since \( f(x) > 0 \) for all \( x eq 0 \).

Step 2: Determine the Domain

The domain \( D \) of \( f \) is all real numbers except where the function is undefined. Since the denominator of \( f(x) = \frac{1}{x^2} \) is zero when \( x = 0 \), the domain is \( \mathbb{R} \setminus \{0\} \), or all real numbers except zero.

Step 3: Determine the Range

The range of \( f(x) = \frac{1}{x^2} \) consists of all possible output values. As \( x^2 > 0 \) for \( x eq 0 \), \( \frac{1}{x^2} \) is always positive and can be very large (as \( x \) approaches 0) or approach 0 (as \( x \) approaches ±∞). Thus, the range \( R \) is \( (0, \infty) \).

Step 4: Find Intervals of Increase or Decrease

To find where \( f(x) \) is increasing or decreasing, check the derivative \( f'(x) \). Compute \( f'(x) = -\frac{2}{x^3} \). This derivative is negative for \( x > 0 \) and positive for \( x < 0 \), meaning \( f(x) \) is decreasing on \( (0, \infty) \) and increasing on \( (-\infty, 0) \).

Key Concepts

Domain and Range

When analyzing functions, it's important to understand two fundamental ideas: domain and range.
The **domain** of a function is the set of all possible input values. For the function \( f(x) = \frac{1}{x^2} \), we encounter a challenge when \( x \) equals zero because division by zero is undefined. As a result, the domain excludes zero and is given as all real numbers except zero, noted as \( \mathbb{R} \setminus \{0\} \).
On the other hand, the **range** consists of all potential output values. For \( f(x) = \frac{1}{x^2} \), regardless of the input (as long as it's not zero), the output will always be positive because a square is always positive. The values of \( f(x) \) can get arbitrarily close to zero but never actually reach it, and they can increase without bound as \( x \) approaches zero from either side. Thus, the range is \( (0, \infty) \), meaning all positive numbers.

Graphing Functions

When we graph \( f(x) = \frac{1}{x^2} \), we see several key characteristics:
This function is a reciprocal quadratic function, meaning it doesn't intersect the x or y-axis and has specific asymptotic behavior.

• **Vertical Asymptote**: Occurs at \( x = 0 \) since the function is undefined there. This axis acts as a boundary that the graph approaches but never touches or crosses.
• **Horizontal Asymptote**: As \( x \) moves toward positive or negative infinity, \( f(x) \) approaches 0. It comes very close to the x-axis without ever crossing it, showcasing another boundary behavior.
• **Symmetry**: The graph is symmetric about the y-axis, which reflects the characteristic of even functions, meaning \( f(x) = f(-x) \) for all \( x \) in the domain. The graph visually verifies the analytical properties of the function.

Increasing and Decreasing Intervals

To determine the intervals where a function is increasing or decreasing, we look at its derivative.
The derivative \( f'(x) = -\frac{2}{x^3} \) tells us the rate of change of \( f(x) \). The sign of this derivative indicates whether \( f(x) \) is increasing or decreasing at any specific interval:

• **Interval \((-\infty, 0)\)**: Here, the derivative is positive. This means that the function \( f(x) \) is increasing in this interval. As \( x \) moves toward 0 from the left, the values of \( f(x) \) become larger.
• **Interval \((0, \infty)\)**: Conversely, in this interval, the derivative is negative, indicating that \( f(x) \) is decreasing. As \( x \) moves away from 0 towards positive infinity, \( f(x) \) steadily decreases.

Understanding these intervals is crucial for analyzing the behavior of the function across its domain.