Problem 12 - Thomas Calculus

Question

a. Graph the function \(f(x)=1 / x .\) What symmetry does the graph have? b. Show that \(f\) is its own inverse.

Step-by-Step Solution

Answer

a. The graph is symmetric about the origin. b. The function is its own inverse: \( f(f(x)) = x \).

Step 1: Identify the Function

The function given is \( f(x) = \frac{1}{x} \). This is a reciprocal function.

Step 2: Graph the Function

To graph \( f(x) = \frac{1}{x} \), notice that the function is undefined at \( x = 0 \), which means there's a vertical asymptote at \( x = 0 \). There is also a horizontal asymptote at \( y = 0 \). As \( x \) approaches positive or negative infinity, \( f(x) \) approaches zero. The graph will be in two parts, one in the first quadrant and the other in the third quadrant.

Step 3: Determine Symmetry

A function is symmetric about the origin if \( f(-x) = -f(x) \). Calculate \( f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x) \). Thus, the graph of \( f(x) = \frac{1}{x} \) is symmetric about the origin.

Step 4: Define the Inverse Function

An inverse function \( f^{-1}(x) \) is one that, when applied to \( f(x) \), gives \( x \) back. For \( y = \frac{1}{x} \), to find the inverse, swap \( x \) and \( y \) to get \( x = \frac{1}{y} \). Solving for \( y \), we get \( y = \frac{1}{x} \). Thus, \( f^{-1}(x) = \frac{1}{x} \).

Step 5: Verify Own Inverse Property

To verify \( f \) is its own inverse, check if applying \( f \) twice returns the input: \( f(f(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right)} = x \). Therefore, \( f(x) = \frac{1}{x} \) is indeed its own inverse.

Key Concepts

Graphing functions

Graphing functions can help us understand how a function behaves and see its important characteristics. Let's look at the function \( f(x) = \frac{1}{x} \), which is a reciprocal function. To graph this function, begin by noticing how it behaves around certain points.

• The function is undefined at \( x = 0 \), which means there is a vertical asymptote there. It shoots up to infinity or down to negative infinity as it approaches zero.
• For very large positive or negative values of \( x \), the function \( f(x) \) gets closer to zero. This indicates a horizontal asymptote at \( y = 0 \).

The graph is split into two distinct parts. One part resides in the first quadrant where both \( x \) and \( y \) are positive, and the other in the third quadrant where both \( x \) and \( y \) are negative. This separation is due to the asymptotes, which the graph approaches but never actually meets. Graphing these features offers a visual representation crucial for understanding and solving many mathematical problems.

Symmetry in functions

A function can exhibit different types of symmetry, and recognizing it can simplify both the analysis and graphing of the function. The function \( f(x) = \frac{1}{x} \), for instance, exhibits symmetry about the origin. Here's why:

• A function is symmetric about the origin if replacing \( x \) with \( -x \) results in \( -f(x) \). Mathematically, this condition can be written as \( f(-x) = -f(x) \).
• For our function, substituting \( -x \) into \( f(x) \) gives \( f(-x) = \frac{1}{-x} = -\frac{1}{x} = -f(x) \).

This result confirms that the graph of \( f(x) = \frac{1}{x} \) has rotational symmetry about the origin. Such symmetry means that if you rotate the graph 180 degrees around the origin, it would look the same. Symmetry can often help make graphing tasks easier and confirm that your work is correct.

Reciprocal function

A reciprocal function like \( f(x) = \frac{1}{x} \) has unique characteristics worth exploring. This function has a fascinating property where it essentially reverses the roles of the numerator and the denominator of a fraction.

• One key feature of the reciprocal function is its domain and range. The domain is all real numbers except zero, since division by zero is undefined. Similarly, the range is all real numbers except zero.
• The reciprocal function also serves as its own inverse. To understand this, switch \( x \) and \( y \) in the equation \( y = \frac{1}{x} \), which leads you to \( x = \frac{1}{y} \). Solving for \( y \), you find \( y = \frac{1}{x} \), demonstrating that \( f^{-1}(x) = f(x) \).
• The fact that \( f(f(x)) = x \) for the function \( f(x) = \frac{1}{x} \) highlights its property as its own inverse, which is quite rare among functions.

Understanding these aspects of the reciprocal function can provide deeper insights into its behavior and how it relates to various mathematical concepts.