91Ó°ÊÓ

The cosine function, often denoted as \( \cos(x) \), is one of the fundamental trigonometric functions. It's always important to understand the range of the cosine function before diving into solving related problems. The range of the cosine function is \([-1, 1]\), which means that no matter what angle or real number \( x \) you choose, \( \cos(x) \) will give you a value between -1 and 1 inclusive.

This property is due to the wave-like behavior of the cosine function as it oscillates between these two bounds. This oscillation occurs because the cosine function is defined on the unit circle, where it represents the x-coordinate of a point at a given angle.

• A minimum value of -1 occurs at angles like \( \pi \) (180 degrees).
• A maximum value of 1 occurs at angles like 0 and \( 2\pi \) (0 and 360 degrees).

Knowing this range is crucial because it dictates the possible values that any expressions involving \( \cos(x) \) can take, especially in inequalities.

Exponential functions, especially those of the form \( e^x \), exhibit several key properties that make them unique and powerful in algebra. One key property is their strictly increasing nature, which means that as the input \( x \) increases, the output \( e^x \) also increases. This property is pivotal in solving inequalities involving exponential functions.

This increasing behavior implies a few things:

• If \( x_1 < x_2 \), then \( e^{x_1} < e^{x_2} \).
• The base \( e \) (approximately 2.71828) is Euler's number, which is the natural base for exponential growth, making it particularly significant in mathematical modeling.
• Even small changes in \( x \) can lead to large changes in \( e^x \), due to the compounded nature of exponential growth.

Thus, when considering \( e^{\cos x} \), the range of \( \cos x \) translates directly into possible outcomes for \( e^x \). Evaluated at \( x=-1 \) and \( x=1 \), the exponential function will vary between approximately 0.367 and 2.718. This property ensures that \( e^{\cos x} \) remains smaller than 3 for all real numbers \( x \).

Solving inequalities involves understanding the relationship between the expressions on each side of the inequality symbol. In the context of our problem, we're dealing with an inequality involving an exponential function \( e^{\cos x} \).

Using what we know about the range of cosine and the properties of exponential functions, we form the inequality: \( e^{-1} \le e^{\cos x} \le e^1 \).

To solve this:

• We first recognize the importance of maintaining the order of the inequality when evaluating exponential functions because \( e^x \) is strictly increasing.
• Evaluate the endpoints of the range of \( \cos x \):
  • \( e^{-1} \approx 0.367 \), which is the minimum possible value of \( e^{\cos x} \) when \( \cos x = -1 \).
  • \( e^{1} = e \approx 2.718 \), which is the maximum value when \( \cos x = 1 \).

From here, it's clear that \( e^{\cos x} \) cannot reach or exceed 3 for any real \( x \). This highlights how leveraging the properties of functions and their ranges enables the effective solving of inequalities.