Problem 2

Question

Examine the graph of $f(x)=\sec x$ on the interval $[-\pi, \pi] .$ How can we tell whether the function is even or odd by only observing the graph of $f(x)=\sec x ?$

Step-by-Step Solution

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Answer

The function $f(x)=\sec x$ is even because its graph is symmetric about the y-axis.

1Step 1: Define Even and Odd Functions

A function is even if for every point $x$ in its domain, $f(x) = f(-x)$. This means the graph is symmetric about the y-axis. A function is odd if for every point $x$, $f(x) = -f(-x)$. This implies rotational symmetry about the origin.

2Step 2: Analyze Symmetry of the Graph

Observe the graph of $f(x) = \sec x$ over the interval $[-\pi, \pi]$. Check for visual symmetry about the y-axis. If the graph mirrors itself across the y-axis, the function is even. If it shows rotational symmetry about the origin, it is odd. Note the vertical asymptotes at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$, which indicate changes in direction.

3Step 3: Test Symmetry Mathematically

Using the properties of the cosine function and the definition $\sec x = \frac{1}{\cos x}$, we know that $\cos(-x) = \cos x$. Thus, $\sec(-x) = \frac{1}{\cos(-x)} = \frac{1}{\cos x} = \sec x$, indicating that $\sec x$ is even.

Key Concepts

SymmetryGraph AnalysisTrigonometric Functions

Symmetry

In mathematics, symmetry is a key characteristic that can help us understand functions better. Let's break it down into two types: even and odd functions.

Even functions are symmetric about the y-axis. This means that if you were to fold the graph along the y-axis, both halves would overlap perfectly. Mathematically, an even function satisfies the condition $f(x) = f(-x)$.
Odd functions, on the other hand, exhibit rotational symmetry about the origin. If you rotate the graph 180 degrees around the origin, it looks the same. For odd functions, the relationship is $f(x) = -f(-x)$.

To determine symmetry from a graph of $f(x) = \sec x$, you can look for these features. Since $\sec x$ is derived from $\cos x$, and we know $\cos(-x) = \cos x$, it makes $f(x)$ an even function. This is evident in its symmetry about the y-axis.

Graph Analysis

Graph analysis involves observing key features of a function's graph to draw conclusions about its properties. When analyzing $f(x) = \sec x$, pay attention to the following aspects:

Symmetry: Identifying whether the function is even or odd is a primary step. As established, $\sec x$ is even due to its y-axis symmetry.
Vertical Asymptotes: $\sec x$ has vertical asymptotes where the cosine of the angle is zero. In the interval $[-\pi, \pi]$, these occur at $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$. These are points where the function "jumps" to positive or negative infinity.
Intervals of Increase and Decrease: Between vertical asymptotes, $\sec x$ behaves predictably—rising or falling steadily.

These analyses help visualize the nature of $\sec x$ and lead to deeper understanding, especially when considering transformations or evaluating behavior over different intervals.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, describing periodic phenomena. $\sec x$, the secant function, is less commonly discussed than sine and cosine but is crucial in complete trigonometric understanding.

Definition and Relation: $\sec x$ is defined as $\frac{1}{\cos x}$, meaning it represents the reciprocal of cosine. This relationship binds $\sec x$'s behavior directly to $\cos x$.
Graph Characteristics: The graph of $\sec x$ features periodic waves and vertical asymptotes corresponding to points where $\cos x = 0$, as these make $\sec x$ undefined.
Applications: Beyond pure mathematics, trigonometric functions including $\sec x$ apply to physics, engineering, and astronomy, modeling phenomena like sound waves or the motion of pendulums.

Understanding each of these aspects in relation to trigonometric functions fosters greater comprehension across various disciplines.

Problem 1

Problem 3

Other exercises in this chapter

Problem 1

We know $g(x)=\cos x$ is an even function, and $f(x)=\sin x$ and $h(x)=\tan x$ are odd functions. What about $G(x)=\cos ^{2} x, F(x)=\sin ^{2} x, $ and

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Problem 3

After examining the reciprocal identity for sec $t,$ explain why the function is undefined at certain points.

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Problem 4

All of the Pythagorean identities are related. Describe how to manipulate the equations to get from $\sin ^{2} t+\cos ^{2} t=1$ to the other forms.

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Problem 5

Use the fundamental identities to fully simplify the expression. $$\sin x \cos x \sec x$$

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