Problem 1 - 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. $$ f(x)=\frac{4}{x} $$

Step-by-Step Solution

Answer

Domain: \((-\infty, 0) \cup (0, \infty)\); Range: \((-\infty, 0) \cup (0, \infty)\); Decreasing on \((0, \infty)\), Increasing on \((-\infty, 0)\).

Step 1: Sketching the Graph

To sketch the graph of \( f(x) = \frac{4}{x} \), recognize that it is a rational function with a vertical asymptote at \( x = 0 \) (since \( x eq 0 \)) and a horizontal asymptote at \( y = 0 \). The graph is hyperbolic, residing in the first and third quadrants, as \( f(x) \) is positive for \( x > 0 \) and negative for \( x < 0 \).

Step 2: Finding the Domain

The domain \( D \) of \( f(x) = \frac{4}{x} \) includes all real numbers except where the function is undefined. The function is undefined at \( x = 0 \), so the domain is \( D = (-\infty, 0) \cup (0, \infty) \).

Step 3: Determining the Range

Since \( f(x) \) approaches infinity as \( x \to 0^+ \) and negative infinity as \( x \to 0^- \), and it can take all values except \( y = 0 \) (due to the horizontal asymptote), the range \( R \) is \( (-\infty, 0) \cup (0, \infty) \).

Step 4: Analyzing Intervals of Increase/Decrease

To determine intervals of increasing or decreasing, examine the derivative \( f'(x) = -\frac{4}{x^2} \). For \( x > 0 \), \( f'(x) < 0 \), indicating that \( f(x) \) is decreasing. For \( x < 0 \), \( f'(x) > 0 \), so \( f(x) \) is increasing. Hence, \( f(x) \) is decreasing on \((0, \infty)\) and increasing on \((-\infty, 0)\).

Key Concepts

Domain and Range

In mathematics, especially when dealing with rational functions like \( f(x) = \frac{4}{x} \), understanding the domain and range is crucial.The **domain** is the set of all possible input values \( x \) that the function can accept without resulting in an undefined expression. For our function, \( f(x) = \frac{4}{x} \), any non-zero value of \( x \) is valid. Because if \( x = 0 \), the function would become undefined. Therefore, the domain of this function is all real numbers except zero:

• Domain: \( D = (-\infty, 0) \cup (0, \infty) \)

The **range** is the set of all possible output values \( y \) that the function can produce.Since \( f(x) = \frac{4}{x} \) can get arbitrarily large or small (negative) as \( x \) approaches zero from either side, it means it can achieve all values except zero itself due to the horizontal asymptote.Therefore, the range is similar to the domain in terms of excluding only the zero:

• Range: \( R = (-\infty, 0) \cup (0, \infty) \)

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches.They play a vital role in shaping the behavior of rational functions like\( f(x) = \frac{4}{x} \). There are two main types of asymptotes to consider:

Vertical Asymptotes

These occur where the function becomes undefined. For \( f(x) = \frac{4}{x} \), data shows a vertical asymptote at \( x = 0 \). This means that as you approach \( x = 0 \) from either direction, the value of \( f(x) \) tends to infinity or negative infinity.

Horizontal Asymptotes

These are determined by the behavior of the function as \( x \to \infty \) or \( x \to -\infty \).For our function, there is a horizontal asymptote at \( y = 0 \).In essence, this means no matter how large or small \( x \) gets, the value of \( f(x) \) will never actually be zero.This creates a situation where the output values endlessly approach zero but never become exactly zero.

Increasing and Decreasing Intervals

Identifying where a function increases or decreases helps in understanding its graph's overall shape.For the function \( f(x) = \frac{4}{x} \), we need to look at its derivative, \( f'(x) = -\frac{4}{x^2} \), to find these intervals. The derivative tells us the slope of the tangent to the graph at any point.

• When \( x > 0 \), the derivative \( f'(x) \) is negative (since both the numerator \(-4\) and \(x^2\) are positive), indicating that the function is decreasing.
• When \( x < 0 \), \( f'(x) \) is also negative because \( -\frac{4}{x^2} \) still evaluates to a positive result (as \( x^2 \) is always positive for non-zero \( x \)). Thus, the function increases.

This analysis leads us to define the intervals:

• **Decreasing:** \( (0, \infty) \)
• **Increasing:** \( (-\infty, 0) \)

By observing these intervals, we can also understand that the graph makes a transition from decreasing to increasing as it moves from the positive to the negative x-axis, which helps in sketching an accurate graph.