Problem 13 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The domain of the function \(h(x)\) is \(-\infty < x < 2\) or \(2 < x < \infty\). There is a vertical asymptote at \(x = 2\), and a horizontal asymptote at \(y = 0\).

Step 1: Finding the Domain of the Function

The domain of a function is all the possible values for \(x\) that will output real numbers. In this rational function, the denominator shouldn’t be 0 because it will make the function undefined. So, set the denominator equal to zero and solve for \(x\): \[(x-2)^{2} = 0\] Solving this gives \(x = 2\). So, the domain of the function \(h(x)\) is all real numbers except 2. Mathematically, the domain is \(-\infty < x < 2\) or \(2 < x < \infty\).

Step 2: Finding the Vertical Asymptotes

The vertical asymptotes of a function are the values of \(x\) that make the denominator equal to zero. Here, the denominator is \((x-2)^{2}\), and setting it to zero gives \(x = 2\). So, the function \(h(x)\) has a vertical asymptote at \(x=2\).

Step 3: Finding the Horizontal Asymptotes

The horizontal asymptotes of a function are found by examining the limits of the function as \(x\) goes to positive or negative infinity. For function \(h(x)\), as \(x\) approaches positive or negative infinity, the value of the function becomes closer and closer to 0. That is, \[\lim_{x \to \pm\infty} \frac{1}{(x-2)^{2}} = 0\] Therefore, there is a horizontal asymptote at \(y = 0\).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (typically denoted as \(x\)) for which the function is defined. For rational functions, which are fractions with polynomials in the numerator and denominator, a key consideration is the values that make the denominator zero. When the denominator is zero, the function becomes undefined because division by zero is not possible.

To determine the domain of the function \(h(x) = \frac{1}{(x-2)^2}\), the first step is setting the denominator equal to zero and solving for \(x\). Here, the equation is \((x-2)^2 = 0\). Solving this yields \(x = 2\). Consequently, \(x\) cannot be 2, since it would make the function undefined. Therefore, the domain of \(h(x)\) is all real numbers except \(x = 2\), which can be expressed in interval notation as \((-\infty, 2) \cup (2, \infty)\).

By understanding the domain, you can predict where the function "lives" and ensure that you work within an interval where the function returns real numbers.

Vertical Asymptotes

Vertical asymptotes are lines that \(x\) approaches but never touches as the function grows indefinitely large or small. These occur where the function is undefined, typically when the denominator equals zero in a rational function, but only if the numerator does not become zero as well at that \(x\) value.

In the case of \(h(x) = \frac{1}{(x-2)^2}\), as deduced from finding the domain, \(x = 2\) causes the denominator to be zero, but the numerator (which is 1) is not zero. Thus, \(x = 2\) is a vertical asymptote. This means as \(x\) approaches 2, the function \(h(x)\) grows infinitely large or small, depending on the direction from which \(x\) approaches.

Remember, vertical asymptotes are key features in sketching the graph of a rational function as they indicate "breaks" or "gaps" in the graph where the function doesn't exist. This helps visualize how the function behaves near these critical points.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \(x\) approaches positive or negative infinity. They can be thought of as "y-values" that the function approaches but might never reach.

For the rational function \(h(x) = \frac{1}{(x-2)^2}\), we analyze the end behavior as \(x\) goes towards infinity in either direction. Specifically, you look for the limit of the function as \(x\) approaches infinity and negative infinity. Here, \[ \lim_{x \to \pm \infty} \frac{1}{(x-2)^2} = 0 \] As \(x\) grows, \(x-2\) also grows, making \((x-2)^2\) even larger, which means \(\frac{1}{(x-2)^2}\) becomes really small, approaching zero. Therefore, there is a horizontal asymptote at \(y = 0\).

Horizontal asymptotes help us understand the long-term trend of rational functions, providing a clear picture of how functions behave as inputs become very large or very small.