Problem 57 - Precalculus: A Concise Course

Question

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ f(x)=\sec x $$

Step-by-Step Solution

Answer

The function \(f(x) = \sec x\) is an even function, but not an odd function.

Step 1: Initial Analysis

The function given is \(f(x) = \sec x\). Now we will derive equations for \(f(-x)\) and \(-f(x)\) to check against \(f(x)\). If \(f(x) = f(-x)\), the function is even and if \(f(x) = -f(-x)\), the function is odd.

Step 2: Checking for evenness

We find the value of \(f(-x)\). The function is \(f(x) = \sec x\), so \(f(-x) = \sec(-x)\). But \(\sec(-x) = 1 / \cos(-x)\). By the rule of cosine, \(\cos(x) = \cos(-x)\), so \(\sec(-x) = \sec(x)\). Here \(f(-x) = f(x)\). So, it's an even function.

Step 3: Checking if the function is also odd

We find the value of \(-f(x)\), which is \(-\sec x\). The value of \(-f(-x)\) is \(-\sec(-x)\), which simplifies to \(-1 / \cos(-x)\). But from the rule of cosine, \(\cos(-x) = \cos(x)\), so \(-\sec(-x) = -\sec(x)\). Here, \(-f(-x) = -f(x)\), which is not equivalent to \(f(x)\). So, \(f(x)\) is not an odd function.

Key Concepts

Secant Function

The secant function, denoted as \(\sec x\), is the reciprocal of the cosine function. This means \(\sec x = \frac{1}{\cos x}\). It plays a crucial role in trigonometry and has unique properties that can be observed in its graph.

Unlike sine and cosine, which are bounded between -1 and 1, the secant function can take on values greater than 1 and less than -1. It has vertical asymptotes wherever \(\cos x = 0\), because division by zero is undefined. These asymptotes occur at intervals \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.

Understanding the secant function is key to analyzing its graph, including identifying regions of increase and decrease, and locating its symmetry.

Graph Analysis

Graph analysis involves examining the shape and features of a function's graph. In the case of \(f(x) = \sec x\), the graph has several interesting characteristics:

• Vertical Asymptotes: As mentioned, these occur where \(\cos x = 0\).
• Periodicity: The secant function is periodic with a period of \(2\pi\), which means the graph repeats every \(2\pi\) units.
• Amplitude: Unlike sine and cosine, the secant's graph doesn't have a maximum amplitude but instead extends to positive and negative infinity where it approaches its asymptotes.

When analyzing this graph, observe how the curves appear like flipped parabolas positioned between the vertical asymptotes. The function is continuous everywhere within its domain, except at the points where it is undefined.

Function Symmetry

Function symmetry is an essential concept to determine if a function is even, odd, or neither. A function is even if \(f(x) = f(-x)\), and it is odd if \(f(x) = -f(-x)\).

By examining \(f(x) = \sec x\), we can determine its symmetry:

• Even Function: Compute \(f(-x)\), which results in \(\sec(-x) = \sec x\). This shows \(f(x) = f(-x)\), confirming that the secant function is even.
• Odd Function: We also compute \(-f(x)\), which leads to \(-\sec x\). Since \(-f(-x)\) simplifies to \(-\sec x\), which is not equal to \(f(x)\), the function is not odd.

Therefore, the secant function exhibits even symmetry, reflecting symmetrical behavior across the y-axis on a graph.