Problem 15 - Precalculus : Building Concepts and Connections

Question

Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{3 x^{2}}{x+1}$$

Step-by-Step Solution

Answer

The function \(h(x) = \frac{3x^2}{x+1}\) is defined for all real numbers except -1. Thus the domain is \(-\infty < x < -1\) and \(-1 < x < \infty\). There is a vertical asymptote at \(x = -1\). Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 1: Finding the Domain

To find the domain of the function set the denominator equal to zero and solve for x: \(x+1 = 0\). Solving this gives \(x = -1\). Since the denominator cannot be zero, this value is excluded from the domain. Thus, the domain of \(h(x)\) is all real numbers except -1: \(-\infty < x < -1\) and \(-1 < x < \infty\).

Step 2: Finding the Vertical Asymptote(s)

The denominators of fractions are not defined when they equal zero. So, if we equate the denominator of the function \((x+1)\) to zero, we can find the vertical asymptote(s). Solving \(x+1 = 0\), we get \(x = -1\). Hence the vertical asymptote is \(x=-1\).

Step 3: Finding the Horizontal Asymptote(s)

Horizontal asymptotes are found by comparing the degree of the numerator and the denominator of a rational function. In this case, the degree of the numerator \((3x^2)\) is 2 and the degree of the denominator \((x+1)\) is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Key Concepts

Domain of a Function

The domain of a function refers to all possible input values (or x-values) that make the function valid or defined. When you have a rational function — a fraction where both the numerator and the denominator are polynomials — the domain is determined based on the denominator being non-zero.

In the function given:

• \( h(x) = \frac{3x^2}{x+1} \)
• The denominator is \( x + 1 \).
• We set it equal to zero: \( x + 1 = 0 \).
• Solving, we find \( x = -1 \) is where the function is undefined.

Therefore, all real numbers except \(-1\) are included in the domain. It's expressed as \((-\infty, -1) \cup (-1, \infty)\).

Remember, every function's domain can be different, and it's important to check denominators and square roots while determining them.

Asymptote

Asymptotes are lines that a graph approaches but never touches. Vertical asymptotes occur when the function approaches infinity, typically at points where the denominator goes to zero in rational functions.

For the function \( h(x) = \frac{3x^2}{x+1} \):

• We found that the denominator \( x+1 = 0 \) leads to \( x = -1 \).
• This indicates a vertical asymptote at \( x = -1 \) because the function becomes undefined, leading it to shoot off infinitely.

Horizontal asymptotes, however, depend on the relative degrees of the numerator and denominator polynomials. If the degree of the numerator is higher (as seen here with degrees 2 and 1), there is no horizontal asymptote. Instead, the function's end behavior resembles a polynomial.

Rational Functions

Rational functions are quotients of two polynomial functions. They are notable for their behavior near zeros of the denominator, which often result in vertical asymptotes.

For the rational function \( h(x) = \frac{3x^2}{x+1} \):

• The numerator is \( 3x^2 \) (degree 2), and the denominator is \( x+1 \) (degree 1).
• Because these types of functions can be split into different regions of behavior divided by asymptotes, understanding their graphs involves looking at domains and asymptotes.

These functions can show complex behavior. At times, they can appear to cross asymptotes, specifically horizontal and oblique ones, though this is under special conditions depending on other parts of the function apart from the main term.