91Ó°ÊÓ

Trigonometric functions are fundamental in mathematics. They relate the angles of a triangle to its side lengths. The three primary trigonometric functions are sine (\textsin\theta), cosine (\textcos\theta), and tangent (\texttan\theta). In this exercise, we focus on the tangent function.

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. It is defined as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Knowing this can help you understand how the tangent function behaves in different contexts. For example, when you input an angle into the tangent function, you get a particular ratio as the output. This insight is crucial when solving composite function problems involving trigonometry.

Function composition is applying one function to the result of another function. It's like following a sequence of instructions step-by-step. If you have two functions, say \textf(x) and \textg(x), the composition of \textg and \textf is written as \textg(f(x)).

In other words, you first apply \textf to \textx, then apply \textg to the result. For example, in our exercise, we are given the functions:

• \[ f(x) = \tan(x) \]
• \[ h(x) = 2x \]
• \[ g(x) = x + 3 \]

To find \[ g(h(f(x))) \], you follow these steps:

• First, apply \[ f(x) = \tan(x) \]
• Next, apply \[ h(x) \] to the result, which means \[ h(f(x)) = h(\tan(x)) = 2\tan(x) \]
• Finally, apply \[ g(x) \] to \[ 2\tan(x) \], resulting in \[ g(2\tan(x)) = 2\tan(x) + 3 \]

This chained process is crucial for solving composite functions accurately.

Mathematical operations include basic arithmetic functions like addition, subtraction, multiplication, and division, as well as more complex functions like those involving trigonometric or exponential operations. Understanding these operations is essential for manipulating functions, especially when they are composed of one another.

Let's break down the mathematical operations involved in our exercise:

• Application of the trigonometric function \[ \tan(x) \] is the first step, which converts the angle \[ x \] into a ratio.
• The next step involves the multiplication operation in \[ h(x) = 2x \]. Here, we multiply the result of \[ \tan(x) \] by 2, giving us \[ 2\tan(x) \].
• Finally, we use addition in \[ g(x) = x + 3 \]. We add 3 to the product \[ 2\tan(x) \], yielding \[ 2\tan(x) + 3 \].

Each step involves applying a mathematical operation to the result of the previous operation. Mastering these basic operations helps you navigate through more complex mathematical puzzles involving composite functions.