91Ó°ÊÓ

Trigonometric functions are a crucial part of mathematics. They relate angles of a triangle to the lengths of its sides. The primary trigonometric functions include sine, cosine, and tangent.
For this exercise, we use the tangent function, written as \(\tan(x)\). The tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle.
Understanding how to apply \(\tan(x)\) in different scenarios, such as in composite functions, is vital.
Composite functions involve using one function inside another, which brings us to the concept of function composition.

Function composition is a concept where one function is applied to the result of another function. It is written as \((f \circ g)(x)\), which means \(f(g(x))\).
For our exercise, we have three functions: \(f(x) = \tan(x)\), \(g(x) = x + 3\), and \(h(x) = 2x\).
To find \(h(f(g(x)))\), we start by solving the innermost function and work our way outwards. In this case, we first find \(g(x)\), then move on to \(f(g(x))\), and lastly solve \(h(f(g(x)))\).
Working step-by-step ensures we correctly apply each function in sequence, helping us to understand and solve complex problems.

Solving complex mathematical problems requires a systematic approach. Here is a detailed look at the steps for our exercise:

Step 1: Find the innermost function
We start with the innermost function \(g(x)\), which is given as \(g(x) = x + 3\). This is our first substitution.

Step 2: Evaluate \(f(g(x))\)
Next, we use the result from step 1 and plug it into the function \(f(x)\), which is \(\tan(x)\). Therefore, \(f(g(x)) = \tan(x + 3)\).

Step 3: Evaluate \(h(f(g(x)))\)
Finally, we substitute \(f(g(x)) = \tan(x + 3)\) into the function \(h(x) = 2x\). Thus, \(h(f(g(x))) = h(\tan(x + 3)) = 2\tan(x + 3)\).
By following these sequential steps, we obtain \(h(f(g(x))) = 2 \tan(x + 3)\).
Breaking down the problem into manageable parts helps in better understanding and solving such equations effectively.