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Chapter 2: Problem 79

Let \(f(x)=\tan (x), g(x)=x+3,\) and \(h(x)=2 x .\) Find the following. $$ f(g(h(x))) $$

Short Answer

Expert verified
\(f(g(h(x))) = \tan(2x + 3)\)

Step by step solution

01

Interpret the Functions

Identify the given functions: \(f(x) = \tan(x)\), \(g(x) = x + 3\), and \(h(x) = 2x\).
02

Evaluate the Innermost Function

Start by evaluating \(h(x)\): Since \(h(x) = 2x\), substitute \(x\) into \(h(x)\) to get \(h(x) = 2x\).
03

Substitute into the Next Function

Take the result of \(h(x)\) and substitute it into \(g(x)\): \(g(h(x)) = g(2x)\). Since \(g(x) = x + 3\), replace \(x\) with \(2x\), so \(g(2x) = 2x + 3\).
04

Final Substitution into the Outer Function

Now take the result of \(g(h(x))\) and substitute it into \(f(x)\): \(f(g(h(x))) = f(2x + 3)\). Since \(f(x) = \tan(x)\), replace \(x\) with \(2x + 3\), so \(f(2x + 3) = \tan(2x + 3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric functions
In trigonometry, trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are fundamental in geometry, engineering, and physics. In the context of the exercise, we are working with the tangent function: \( f(x) = \tan(x) \). The tangent function represents the ratio of the opposite side to the adjacent side of a right triangle. Understanding tangent and other trig functions is key in trigonometry calculations.
function composition
Function composition involves combining two or more functions where the output of one function becomes the input of the next. Let's break it down using our example from the exercise. We have:
  • \(f(x) = \tan(x)\)
  • \(g(x) = x + 3\)
  • \(h(x) = 2x\)
The goal is to find \(f(g(h(x)))\). This is done in steps:
  • First, evaluate the innermost function: \(h(x) = 2x\).
  • Next, use this result as the input for the second function: \(g(2x) = 2x + 3\).
  • Finally, take the result from the second function and put it into the outer function: \(f(2x + 3) = \tan(2x + 3)\).
The final result combines all three functions into one expression.
step-by-step problem solving
Solving math problems step-by-step helps to simplify complex operations and ensure correctness. Here's how to approach the given problem:

  • Step 1: Identify the functions. We have \(f(x) = \tan(x)\), \(g(x) = x + 3\), and \(h(x) = 2x\).

  • Step 2: Evaluate the innermost function. Calculate \(h(x)\): \(h(x) = 2x\).

  • Step 3: Substitute into the next function. Use the result of \(h(x)\) as input to \(g(x)\): \(g(h(x)) = g(2x)\). Since \(g(x) = x + 3\), substitute to get \(g(2x) = 2x + 3\).

  • Step 4: Final substitution into the outer function. Take \(g(h(x))\) and use it as input for \(f(x)\): \(f(g(h(x))) = f(2x + 3)\). Since \(f(x) = \tan(x)\), replace \(x\) to get \(\tan(2x + 3)\).
Breaking down each step ensures accuracy and understanding. This method can be applied to a variety of similar problems in trigonometry and beyond.

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Most popular questions from this chapter

Find the period and range of the function \(y=\cot (\pi x)\).

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