Problem 37 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{|x|}{x+1}\) is at y = 1.

Step 1: Define the function domains

There are two cases to consider when finding the function value due to the absolute value operation.|x| = x when x>=0 and |x| = -x when x<0. Therefore, f(x) = x/(x+1) when x>=0 and f(x) = -x/(x+1) when x<0.

Step 2: Identify horizontal asymptotes

The horizontal asymptote of a function can be identified from the degree of the polynomial in the numerator and the denominator. If the degree of the polynomial in the numerator is less than or equal to the degree in the denominator, there will be a horizontal asymptote at y = 0. If the degree is greater in the numerator, there is no horizontal asymptote. In both function cases, the degree of the numerator and the denominator is the same, implying a horizontal asymptote at y = the ratio of the leading coefficients, which is 1.

Step 3: Graph the function

Plotting the function involves graphing two separate lines for our two function domains, with the point of discontinuity at x=-1. For x>=0, the line increases from the asymptote at y = 1. For x<0, the line decreases towards the asymptote at y = 1. Consequently, we see that the function approaches y = 1 as x approaches infinity and negative infinity, visually confirming a horizontal asymptote at y = 1.

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visualize the behavior of mathematical expressions. To graph a function, you plot points on a coordinate grid that represent the relationship between variables defined by the function. In the case of the function \(f(x) = \frac{|x|}{x+1}\), graphing involves considering the behavior of the function in different regions. For this function, you split the domain into two parts based on the absolute value of \(x\).

• For \(x \geq 0\), the function behaves as \(f(x) = \frac{x}{x+1}\).
• For \(x < 0\), it behaves as \(f(x) = \frac{-x}{x+1}\).

By plotting points for varying \(x\) values, we see how each section approaches the horizontal asymptote at \(y = 1\), giving us a comprehensive picture of \(f(x)\) as a whole.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input, \(x\), gets very large or very small. They represent stable behavior, showing how the function behaves as \(x\) goes to infinity or negative infinity. To determine a horizontal asymptote, compare the degrees of the highest powers in the numerator and denominator of the function's rational form.

In \(f(x) = \frac{|x|}{x+1}\), adjust for the absolute value and identify the behavior for each case:

• If \(x \geq 0\), it acts like \(\frac{x}{x+1}\).
• If \(x < 0\), it acts like \(\frac{-x}{x+1}\).

In both scenarios, the degree of the numerator and denominator are equal. This equality results in the horizontal asymptote \(y = 1\), derived from the ratio of leading coefficients in both cases.

Absolute Value Functions

Absolute value functions introduce an interesting condition: they are defined by their tendency to always return a non-negative output. For \(f(x) = \frac{|x|}{x+1}\), different definitions apply based on whether \(x\) is positive or negative.

• For positive \(x\), the absolute value has no effect: \(|x| = x\).
• For negative \(x\), the absolute value negates the factor making \(|x| = -x\).

These conditions split the function into two parts, impacting how you graph and analyze it. When dealing with absolute value functions, think about how the function behaves differently in the positive and negative sections of its domain.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of variables with coefficients. They form a fundamental building block in calculus. When considering horizontal asymptotes, understanding the polynomial degrees in the numerator and denominator is crucial.
For the function \(f(x) = \frac{|x|}{x+1}\), when handled piecewise:

• The numerator \(|x|\) changes based on the sign of \(x\).
• The denominator polynomial, \(x+1\), remains constant in degree 1.

In both function segments, examine the power of \(x\). With equivalent degrees (both 1), the horizontal asymptote relates to the ratio of leading coefficients, finalizing at \(y = 1\). The polynomial framework thereby assists with the asymptotic identity determination.