Problem 61 - Precalculus

Question

For the following exercises, find the period and horizontal shift of each of the functions. If \(\tan x=-1.5,\) find \(\tan (-x)\)

Step-by-Step Solution

Answer

\(\tan(-x) = 1.5\)

Step 1: Understanding the Property of Tangent Function

The tangent function is an odd function. This means that for any value of \(x\), the following identity holds: \( \tan(-x) = -\tan(x) \). This identity will help us find \( \tan(-x) \) when \( \tan(x) = -1.5 \).

Step 2: Applying the Odd Function Property

Since \( \tan(x) = -1.5 \), using the property of odd functions, we have:\[\tan(-x) = -\tan(x) = -(-1.5)\]

Step 3: Calculating \(\tan(-x)\)

Perform the arithmetic calculation:\[\tan(-x) = -(-1.5) = 1.5\] Thus, \( \tan(-x) = 1.5 \).

Key Concepts

Odd Function

An odd function is characterized by the property that reflecting it across the origin (or flipping it over both axes) results in the same function value but with an opposite sign. In mathematical terms, a function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain. This property means that the function's graph is symmetric with respect to the origin. Key characteristics of odd functions include:

• Graph symmetry around the origin
• Opposite outputs for opposite inputs
• Examples include \( y = x^3 \) and \( y = \sin(x) \)

This concept is crucial when studying trigonometric functions like tangent, as it allows us to find values like \( \tan(-x) \) by simply changing the sign of \( \tan(x) \). When \( \tan(x) = -1.5 \), this property helps us deduce that \( \tan(-x) = 1.5 \).

Tangent Function

The tangent function, \( \tan(x) \), is one of the primary trigonometric functions. It is defined as the ratio of the sine to the cosine of an angle in a right triangle; mathematically, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Some important properties of the tangent function include:

• It is periodic with a period of \( \pi \), meaning the function repeats its values every \( \pi \) radians.
• The function is undefined wherever the cosine function is zero, leading to vertical asymptotes at \( x = \frac{(2n+1)\pi}{2} \) for integers \( n \).
• Since it is an odd function, the tangent of the negation of any angle is the negation of the tangent of the angle. This is evident in \( \tan(-x) = -\tan(x) \).

By understanding these properties, students can analyze the behavior of the tangent function over its domain and apply identities effectively.

Period of Function

The period of a function is the interval over which it completes one full cycle before repeating its pattern. For example, the classic sine and cosine functions have a period of \( 2\pi \), meaning their graphs repeat every \( 2\pi \) units along the x-axis. In contrast, the tangent function has a shorter period of only \( \pi \). This means that every \( \pi \) radians, the tangent function repeats its cycle. Understanding a function's period is vital when analyzing its graph, predicting its values, or solving equations involving trigonometric functions. For trigonometric functions like tangent, knowing the function's odd nature and its period of \( \pi \) assists in simplifying calculations and finding equal angles that produce similar outputs within this interval.

Horizontal Shift

A horizontal shift in a function involves moving the entire graph left or right along the x-axis. This is a type of transformation that adjusts the position of a function without altering its shape or orientation.When a function \( y = f(x) \) is shifted horizontally by \( c \) units, it becomes \( y = f(x - c) \). This means:

• If \( c \) is positive, the graph shifts to the right.
• If \( c \) is negative, the graph shifts to the left.

In the case of trigonometric functions, horizontal shifts are often used to phase-shift the graph, helping align functions with specific conditions or data points. For example, adjusting the starting point of a periodic function like cosine or sine to model real-world phenomena such as tides or sound waves. Understanding horizontal shifts is crucial in graph transformations, allowing us to manipulate functions effectively for various applications.