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Chapter 3: Problem 37

In Exercises 37-40, use a graphing utility to graph the function and identify any horizontal asymptotes. $$f(x)=\frac{|x|}{x+1}$$

Short Answer

Expert verified
The function \(\frac{|x|}{x+1}\) has a horizontal asymptote at y=1.

Step by step solution

01

Understand the Function

The given function is \(\frac{|x|}{x+1}\), the absolute value of x divided by x plus 1, valid for \(x \neq -1\) as the denominator should not be zero.
02

Graph the Function

Remembering that the absolute value function gives a positive value or zero for any real number, it can be understood that the function will produce two distinct cases for x values. They are \(x < -1\) and \(x > -1\). The graph should be plotted accordingly.
03

Identify Horizontal Asymptotes

Horizontal asymptotes of the graph tell us the value that y approaches as x gets larger or smaller. From the graph, it can be noticed that the horizontal asymptote is y=1 because as x goes to positive or negative infinity, the value of the function \(f(x)\) approaches 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions helps us visualize how a function behaves across different values of x. For the function \( f(x) = \frac{|x|}{x+1} \), it鈥檚 important to consider where the function exists and behaves differently. Since the denominator \( x+1 \) cannot be zero, x cannot be -1. This means we should graph the function focusing on the intervals \( x < -1 \) and \( x > -1 \).

When graphing:
  • Identify the domain where the function is defined.
  • Consider how the function behaves differently across parts of the domain based on the behavior of the numerator \( |x| \).
  • Plot key points, especially around the critical point \( x = -1 \) where there鈥檚 a change in behavior.
Graphing the function effectively involves understanding symmetries and transformations, helping reveal a function's complete behavior.
Absolute Value Function
The absolute value function, \( |x| \), plays a crucial role here. Absolute value functions take any real number and return its positive magnitude. This causes the function \( \frac{|x|}{x+1} \) to split into different behaviors based on the sign of x.

For \( x < 0 \):
  • The absolute value of x \((|x|)\) makes negative x values positive.
  • The graph reflects the points above the x-axis, turning negatives into positives.
For \( x > 0 \):
  • \( |x| \) is simply x since x is already positive.
  • The graph remains unchanged beyond simplifying the denominator.
Understanding how the absolute value impacts the graph is key to predicting the overall shape.
Asymptotic Behavior
Asymptotic behavior is about understanding how a function behaves as x approaches extreme values, like infinity or negative infinity. For \( f(x) = \frac{|x|}{x+1} \), this involves analyzing how the values get closer to a line, known as a horizontal asymptote, without actually touching it.

Horizontal asymptotes can be identified by considering limits:
  • As \( x \to \infty \), \( f(x) \to 1 \).
  • As \( x \to -\infty \), \( f(x) \to 1 \) as well.
This symmetry suggests that \( y = 1 \) is the horizontal asymptote. The significance of asymptotic behavior is that it outlines the end behavior of the graph, helping to understand the function's long-term trends beyond just looking for intercepts and local extremes.

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