Question

Graph the function. State the domain and range. $$ p(x)=\frac{3}{x+1}-2 $$

Short answer

Domain is \( D = (-\infty,-1) \cup (-1,+\infty) \) and the range is \( R = (-\infty,+\infty) \). The function approaches but never reaches the vertical line \( x = -1 \).

Step-by-step solution

Identify the values that make the denominator zero

This is done by setting the expression \(x +1 = 0\). Solving for \(x\) gives \(x = -1\). This value is not included in the domain.

Specify the Domain

As we identified that \( x \neq -1\), the domain of \( p(x) \) is all real numbers except -1 that is, \( D = (-\infty,-1) \cup (-1,+\infty) \).

Find the Range

For a rational function of this form, the range is typically all real numbers. This is also true for this specific function as there are no minimum or maximum y-values. Therefore, the range can be represented as \( R = (-\infty,+\infty) \).

Graph the Function

To graph this function, mark the point (-1,-2) as a vertical asymptote. This means the function approaches but never reaches this value. The function is decreasing for \( x < -1 \) and increasing for \( x > -1 \), so sketch the function accordingly.

Key concepts

Domain and Range

When graphing rational functions, it's imperative to determine the set of input values (domain) and output values (range) the function can take.
For the function \( p(x) = \frac{3}{x+1} - 2 \), the domain consists of all real numbers except where the denominator is zero, because division by zero is undefined.
By solving \( x + 1 = 0 \), we find that \( x = -1 \) is not in the domain, leaving us with the domain: \( D = (-\infty, -1) \cup (-1, +\infty) \).
In terms of range, this function has a range of all real numbers, \( R = (-\infty, +\infty) \), because for any rational function without restrictions on its outputs, any real number can be achieved depending on the input value.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator is zero, but they signify a unique behavior of the function at those points. At \( x = -1 \), the function \( p(x) = \frac{3}{x+1} - 2 \) experiences a vertical asymptote.
This means that as \( x \) approaches \(-1\) from either side, the function's value increases or decreases without bound (moves towards infinity).

• Vertical asymptotes are not part of the function's graph, but they guide its behavior.
• They divide the graph into two sections where the function behaves differently.

Even though the function will never actually touch or cross the vertical asymptote, the graph will get arbitrarily close as \( x \) continues to approach \(-1\) from either side.

Rational Functions

Rational functions are quotients of two polynomials, like \( p(x) = \frac{3}{x+1} - 2 \), where the numerator and denominator are polynomial expressions.
They are pivotal in understanding various properties of functions due to their characteristic asymptotic behaviors.

• The numerator determines the height of the curve, while the denominator determines the function's horizontal and vertical asymptotes.
• A rational function can have holes, vertical asymptotes, and different types of behavior at infinity.

Knowing the basic characteristics of the numerator and the denominator allows us to split the analysis of the entire function into more manageable parts.

Function Graph Analysis

When graphing a rational function like \( p(x) = \frac{3}{x+1} - 2 \), it's beneficial to methodically analyze its key features to draw an accurate graph.
Follow these steps to ensure a comprehensive understanding:

• Identify asymptotes: Vertical asymptotes show where the function value spikes to infinity, here at \( x = -1 \).
• Determine if there are any intercepts: For example, find where the graph crosses the axis by setting \( p(x) \) to zero or solving for any other intercepts.
• Analyze the behavior at infinity: Understand how the function behaves as \( x \) moves towards positive or negative infinity to sketch the ends.
• Consider the function's symmetry: This helps make drawing the graph easier.

By correctly identifying and sketching these features, you can accurately represent the graph of any rational function.