/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Find \(f \circ g \circ h\). $$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 39

Find \(f \circ g \circ h\). $$ f(x)=3 x-2, \quad g(x)=\sin x, \quad h(x)=x^{2} $$

Short Answer

Expert verified
The function \((f \circ g \circ h)(x)\) is \(3\sin(x^2) - 2\).

Step by step solution

01

Identify Functions

We are given three functions: \( f(x) = 3x - 2 \), \( g(x) = \sin x \), and \( h(x) = x^2 \). We need to find the composition of these functions \((f \circ g \circ h)(x)\).
02

Compose Inner Function

Start by finding \( g(h(x)) \). Since \( h(x) = x^2 \), substitute into \( g(x) \) to get \( g(h(x)) = \sin(x^2) \).
03

Compose Outer Function

Next, substitute \( g(h(x)) \) into \( f(x) \) to get \( f(g(h(x))) \). This means substituting \( g(h(x)) = \sin(x^2) \) into \( f(x) = 3x - 2 \) to obtain \( f(\sin(x^2)) = 3\sin(x^2) - 2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mathematical Functions
Mathematical functions are the building blocks of many concepts in mathematics. They associate every element in a domain with exactly one element in a range. In simpler terms, a function is like a "rule" or "machine." You input something (from the domain), and it gives you an output (in the range).
For example, consider the function \( f(x) = 3x - 2 \). Here, for every value of \( x \) you input, the function multiplies it by 3 and then subtracts 2.
  • The domain of a function is the set of all possible inputs.
  • The range is the set of all possible outputs.
Understanding functions is essential for higher mathematical concepts like function composition, where you apply one function to the results of another.
Trigonometric Functions
Trigonometric functions are special mathematical functions related to angles of triangles and the unit circle. They are fundamental in studying periodic phenomena, like waves and cycles. One common trigonometric function is the sine function: \( g(x) = \sin x \).
This function outputs the sine of an angle \( x \), which is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Trigonometric functions are periodic, meaning they repeat their values in regular intervals.
  • Sine, cosine, and tangent are the most well-known trigonometric functions.
  • These functions are essential in modeling periodic processes in nature and engineering.
In the exercise, the sine function \( g(x) \) is composed with other functions to create a more complex mathematical expression, demonstrating its versatility.
Calculus Problem-Solving
Calculus problem-solving often involves understanding how functions change and interact. One common operation within calculus is function composition, where the output of one function becomes the input of another. This allows for the creation of complex expressions through simpler ones.
In our exercise, we have the task of composing three functions: \( f(x) = 3x - 2 \), \( g(x) = \sin x \), and \( h(x) = x^2 \).
  • We start by finding \( g(h(x)) \), meaning we substitute \( h(x) = x^2 \) into \( g(x) = \sin x \), resulting in \( g(h(x)) = \sin(x^2) \).
  • Next, we substitute \( g(h(x)) \) into \( f(x) \) to find \( f(g(h(x))) \). That means replacing \( x \) in \( f(x) = 3x-2 \) with \( \sin(x^2) \), giving us \( f(\sin(x^2)) = 3\sin(x^2) - 2 \).
This composition involves applying multiple layers of transformations, which is a powerful method for solving complex calculus problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Express the given quantity as a single logarithm. $$ \ln b+2 \ln c-3 \ln d $$

Find the domain and range of the function $$ g(x)=\sin ^{-1}(3 x+1) $$

(a) How is the logarithmic function \(y=\log _{b} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function \(y=\log _{b} x\) if \(b>1\).

Find an explicit formula for \(f^{-1}\) and use it to graph \(f^{-1},\) \(f,\) and the line \(y=x\) on the same screen. To check your work, see whether the graphs of \(f\) and \(f^{-1}\) are reflections about the line. $$ f(x)=1+e^{-x} $$

Find a formula for the inverse of the function. $$ f(x)=\frac{4 x-1}{2 x+3} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.