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Chapter 2: Problem 70

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

Short Answer

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

Step by step solution

01

Find h(x)

Given that \( h(x) = 2x \), we need to start by identifying \( h(x) \).
02

Apply h(x) to f(x)

Next, substitute \( h(x) \) into \( f(x) \). Since \( f(x) = \tan(x) \), we need to find \( f(h(x)) \). Therefore, \( f(h(x)) = \tan(2x) \).
03

Apply f(h(x)) to g(x)

Finally, substitute \( f(h(x)) \) into \( g(x) \). Given that \( g(x) = x + 3 \), we need to find \( g(f(h(x))) \). Therefore, \( g(f(h(x))) = g(\tan(2x)) = \tan(2x) + 3 \).

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Key Concepts

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

Function Composition
In mathematics, function composition refers to applying one function to the result of another. Imagine functions as machines. Each machine takes an input, processes it, and gives an output. When we compose functions, the output of one machine becomes the input of another. In mathematical notation, if you have two functions, say \(f(x)\) and \(g(x)\), the composition is denoted as \( g(f(x)) \). This means you first apply \(f\) to \(x\) and then apply \(g\) to the result from \(f(x)\).

Let's break down the composition, \(g(f(h(x)))\), from the example:
  • First, you apply the innermost function: \( h(x) = 2x \).
  • Next, take the result from \( h(x) \) and apply the next function: \( f(2x) = \tan(2x) \).
  • Finally, apply the last function: \( g(\tan(2x)) = \tan(2x) + 3\).
Function composition can initially seem tricky, but breaking it down step-by-step will make it manageable.
Trigonometric Functions
Trigonometric functions are functions related to angles and sides of triangles. The most common ones used are sine (\(\text{sin}\)), cosine (\(\text{cos}\)), and tangent (\(\text{tan}\)). In our example, the function \( f(x) \) is a tangent function, represented as \( \tan(x) \).

Here's a quick overview of the tangent function:
  • The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
  • The graph of \( \tan(x) \) repeats every \(\frac{\text{蟺}}{2}\) radians or 90 degrees, showing a periodic behavior.
  • Unlike sine and cosine, the range of the tangent function is all real numbers, meaning it can take any real value.
In our exercise, when we have \( f(h(x)) \), we calculate \( \tan(2x) \), which indicates the tangent function applied to \(2x\). This transformation takes into account the nature of tangent and how it behaves with different input values.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with. In the context of function composition and trigonometric functions, it often means substituting one function into another and simplifying the resulting expression.

For our specific problem, let's revisit the algebraic steps needed to achieve the final answer:
  • Start with \( h(x) = 2x \).
  • Then find \( f(h(x)) \). Since \( f(x) = \tan(x) \), substituting gives us \( f(2x) = \tan(2x) \).
  • Finally, apply \( g \) to the result from \( f(h(x)) \). Given \( g(x) = x + 3 \), we replace \( x \) by \( \tan(2x) \), obtaining \( g(\tan(2x)) = \tan(2x) + 3 \).
The key is to perform each function step by step and ensure each transformation is correctly applied. Knowing how to manipulate algebraic expressions effectively will help in handling more complex problems in both algebra and trigonometry.

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Most popular questions from this chapter

Describe the graph of each function then graph the function between -2 and 2 using a graphing calculator or computer. $$ y=\frac{1}{x}+\cos \pi x $$

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=\sin (x-\pi / 2)-2 $$

Determine the point that lies midway between the two given points. $$ (3 \pi / 4,5) \text { and }(\pi, 5) $$

Find the equation of each sine wave in its final position. The graph of \(y=\sin (x)\) is reflected in the \(x\) -axis, shrunk by a factor of \(\frac{1}{2},\) shifted \(\pi / 3\) units to the right, and then translated upward 4 units.

For the past three years, the manager of The Toggery Shop has observed that the utility bill reaches a high of about \(\$ 500\) in January and a low of about \(\$ 200\) in July, and the graph of the utility bill looks like a sinusoid. If the months are numbered 1 through 36 with 1 corresponding to January, then what are the period, amplitude, and phase shift for this sinusoid? What is the vertical translation? Write a formula for the curve and find the approximate utility bill for November.
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