Problem 11 - Precalculus : Building Concepts and Connections

Question

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{-x^{2}+9}{-2 x^{2}+8}$$

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty, -2 \sqrt (2)) \cup (-2 \sqrt (2), 2 \sqrt (2)) \cup (2 \sqrt (2), \infty) \). The vertical asymptotes are \( x= \pm 2 \sqrt (2) \) and the horizontal asymptote is \( y=0.5 \).

Step 1: Identifying Domain of the Function

Domain of a function is the set of all real numbers for which the function is defined. For the function \( f(x)=\frac{-x^{2}+9}{-2 x^{2}+8} \), it is defined for all values of \( x \) except those for which the denominator is zero. So, let's find the values of \( x \) for which the denominator is zero: \n\[-2 x^{2}+8=0\]\n Solving for x, we find \( x= \pm 2 \sqrt (2) \). As a result, the domain of the given function is all real numbers except \( \pm 2 \sqrt (2) \). So, the domain is \( (-\infty, -2 \sqrt (2)) \cup (-2 \sqrt (2), 2 \sqrt (2)) \cup (2 \sqrt (2), \infty).

Step 2: Identifying Vertical Asymptotes

Vertical asymptotes can be found by determining the values of \( x \) which make the denominator of the function zero. As we found in Step 1, \( x= \pm 2 \sqrt (2) \) make the denominator zero. So, the function \( f(x) \) has vertical asymptotes at \( x= \pm 2 \sqrt (2) \)

Step 3: Identifying Horizontal Asymptotes

Horizontal asymptotes are determined by finding the limit of the function as \( x \) approaches infinity and negative infinity. Horizontal asymptotes require comparing the highest degrees of the numerator and denominator. If the degrees are the same, then the horizontal asymptote is the ratio of the coefficients of the highest degree terms. Thus in our case, the horizontal asymptote is \( y = \frac{-1}{-2} = 0.5 \)

Key Concepts

Domain of a function

Understanding the domain of a function is essential in precalculus. It's like knowing the boundaries within which a function can operate. In simpler terms, the domain is the set of all possible input values (usually \( x \)) that allow the function to work without any hiccups, like division by zero or taking the square root of a negative number.

For the function \( f(x)=\frac{-x^{2}+9}{-2 x^{2}+8} \), the trick to finding the domain lies in ensuring that the denominator \(-2x^{2}+8\) is never zero because dividing by zero is a no-no in mathematics.

Let's break it down:

• Set the denominator to zero: \(-2x^{2}+8 = 0\)
• Solve for \( x \): \(x= \pm 2 \sqrt{2}\)

This means the function is undefined at these \( x \) values.

Hence, the domain of \( f(x) \) excludes these values, resulting in the domain: \((-\infty, -2 \sqrt{2}) \cup (-2 \sqrt{2}, 2 \sqrt{2}) \cup (2 \sqrt{2}, \infty)\).

Vertical asymptotes

Vertical asymptotes are lines where the function shoots up to infinity (or down to negative infinity). They occur at \( x \)-values that make the denominator zero while the numerator isn't zero at the same time.

For \( f(x)=\frac{-x^{2}+9}{-2x^{2}+8} \), following our domain calculation, we found that the function is not defined at \( x = \pm 2 \sqrt{2} \).

• These \( x \) values cause the denominator to be zero but don't zero out the numerator simultaneously.
• Thus, they are vertical asymptotes, meaning the graph of the function approaches but never touches these vertical lines.

In summary, the vertical asymptotes are at \( x = \pm 2 \sqrt{2} \). The graph approaches these lines, creating sharp turns or breaks.

Horizontal asymptotes

Horizontal asymptotes describe the behavior of a function as it heads toward infinity or negative infinity.

When evaluating horizontal asymptotes of the function \( f(x)=\frac{-x^{2}+9}{-2x^{2}+8} \), we compare the highest power of \( x \) in the numerator and denominator:

• The highest power in both the numerator and the denominator is \( x^2 \).
• The leading coefficients are \( -1 \) in the numerator and \( -2 \) in the denominator.

To find the horizontal asymptote, we simply take the ratio of these coefficients:

\[ y = \frac{-1}{-2} = 0.5 \] This means that as \( x \) moves toward positive or negative infinity, \( f(x) \) approaches the line \( y = 0.5 \).

Horizontal asymptotes give us insight into the long-term behavior of functions, helping us predict where graphs will level off.