Question

Specify whether the given function is even, odd, or neither, and then sketch its graph. $$ g(x)=\frac{x}{x^{2}-1} $$

Short answer

The function is odd; sketch with vertical asymptotes at x = ±1.

Step-by-step solution

Understand function types

A function is considered even if for every x in the function's domain, \( f(-x) = f(x) \). A function is odd if for every x, \( f(-x) = -f(x) \). If neither condition is satisfied, the function is neither even nor odd.

Substitute -x into the function

To check for evenness or oddness, substitute \(-x\) into the given function \( g(x) = \frac{x}{x^2 - 1} \). The result is \( g(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} \).

Compare g(-x) with g(x)

Compare \( g(-x) = \frac{-x}{x^2 - 1} \) with \( g(x) = \frac{x}{x^2 - 1} \). Here, \( g(-x) = -g(x) \), which satisfies the condition for the function being odd.

Conclude about function type

Since \( g(-x) = -g(x) \) holds true for the domain of the function, \( g(x) = \frac{x}{x^2 - 1} \) is an odd function.

Identify discontinuities for sketching

Identify where the function is undefined. The function has vertical asymptotes when \( x^2 - 1 = 0 \), thus \( x = \pm 1 \). These are the points of discontinuity.

Sketch the graph of the function

Draw the graph of \( g(x) \) showing symmetry about the origin due to oddness. Indicate vertical asymptotes at \( x = 1 \) and \( x = -1 \). The graph approaches zero as x moves away from the origin.

Key concepts

Function Analysis

Function analysis involves examining and breaking down a function to understand its properties and behavior. When analyzing a function, we look at several aspects including:

• The domain and range: Understanding where the function is defined and the outputs it produces.
• The behavior at infinity: Observing how the function behaves as the input grows very large or very small.
• The function type: Determining if it's polynomial, rational, exponential, etc.

For the given function, \[ g(x) = \frac{x}{x^2 - 1} \]we start by noting that it's a rational function, composed of a polynomial numerator and a polynomial denominator.
The domain is restricted by the denominator; it cannot be zero, leading to points where the function is undefined.
Analyzing these aspects helps in understanding the full scope of the function's behavior.

Symmetry in Functions

Symmetry in functions can greatly simplify the process of graphing by indicating a repeatable pattern. When a function exhibits symmetry, it means:

• For even functions: The function is symmetric about the y-axis, meaning \( f(-x) = f(x) \).
• For odd functions: The function is symmetric around the origin, meaning \( f(-x) = -f(x) \).

For our function \[ g(x) = \frac{x}{x^2 - 1} \], substituting \(-x\) for \(x\) results in \[ g(-x) = \frac{-x}{x^2 - 1} = -g(x) \].
This confirms that \( g(x) \) is an odd function, so it possesses origin symmetry.
Recognizing this allows us to sketch the graph more accurately, ensuring it reflects this symmetry.

Vertical Asymptotes

Vertical asymptotes are lines where a function's value approaches infinity, and they often indicate points of discontinuity. They occur in rational functions when the denominator approaches zero:

• Vertical asymptotes indicate where the function is not defined.
• The graph approaches the asymptote but never actually touches it, displaying a steep incline or decline.

For our function \[ g(x) = \frac{x}{x^2 - 1} \], set the denominator to zero: \( x^2 - 1 = 0 \).Solving this gives \( x = \pm 1 \).
Each of these values represents a vertical asymptote. When sketching, it's important to draw dashed lines at these points to indicate that the function is undefined here, marking the sharp boundaries of the graph.

Discontinuities in Functions

Discontinuities are points where a function is not continuous, often disrupting the graph. They are critical to identify as they mark where the function line breaks:

• They typically occur at vertical asymptotes or when a function involves division by zero.
• At these points, the graph will not connect smoothly, and the function does not have a value.

For the function \[ g(x) = \frac{x}{x^2 - 1} \], the discontinuities are located where its denominator equals zero, specifically at \( x = 1 \) and \( x = -1 \).
These are the locations of the vertical asymptotes, which align with the characteristic places for discontinuities.
Understanding where these occur helps in accurately depicting the function's graph and understanding its overall behavior.