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Chapter 1: Problem 7

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=x+1, \quad g(x)=3 x, \quad h(x)=4-x $$

Short Answer

Expert verified
The formula for \((f \circ g \circ h)(x)\) is \(13 - 3x\).

Step by step solution

01

Understand the Problem

We need to find the formula for the composition \((f \circ g \circ h)(x)\), which means we need to apply the functions \(f\), \(g\), and \(h\) in order from the innermost to the outermost function.
02

Apply Function h

The innermost function is \(h(x) = 4 - x\). Apply \(h\) to \(x\):\[ h(x) = 4 - x \]
03

Apply Function g

Next, apply function \(g\) to the result of \(h(x)\). Substitute \(h(x)\) into \(g(x) = 3x\):\[ g(h(x)) = g(4-x) = 3(4-x) = 12 - 3x \]
04

Apply Function f

Finally, apply function \(f\) to the result of \(g(h(x))\). Substitute \(g(h(x))\) into \(f(x) = x + 1\):\[ f(g(h(x))) = f(12 - 3x) = (12 - 3x) + 1 = 13 - 3x \]
05

Write the Formula for f \, \circ \, g \, \circ \, h

Combining all the steps, \((f \circ g \circ h)(x)\) is given by:\[ (f \circ g \circ h)(x) = 13 - 3x \]

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Key Concepts

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

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a way to represent numbers in equations and expressions. In this problem, algebra helps us to see how functions change the input values and transform them into different outputs.
  • In our example, we have three functions: f(x) = x + 1, g(x) = 3x, and h(x) = 4 - x.
  • Each of these functions represents a specific algebraic operation.
Using algebra, we combine these functions in sequence, or "compose" them, leading us to a final expression by substituting one function into another. The composition requires applying each function step by step to transform the original variable, illustrating the power of algebraic manipulation. This method enables us to solve complex transformations and operations one step at a time.
Function Notation
Function notation is a way of expressing functions using symbols. It tells us how to input a value into a function and what operation will be performed.
  • The notation f(x) tells us that f is a function that involves the variable x.
  • Similarly, g(x) and h(x) use this notation to represent the operations of those functions.
Function notation lets us clearly express complex operations, especially when dealing with multiple functions. In this exercise, it helps us communicate each step of the function composition clearly. We use the notation to substitute results from one function as the input for another, ultimately leading us to the composed function \((x) = 13 - 3x\).Function notation not only simplifies expressing calculations but also provides a way to visualize and track how values change through operations.
Step-by-step Solution
A step-by-step solution is essential for understanding function composition. It breaks down the complex process into manageable parts, which simplifies learning.
  • **Step 1:** Recognize the problem as a composition of functions.
  • **Step 2:** Apply the innermost function, starting with h(x) = 4 - x.
  • **Step 3:** Use the result from Step 2 as an input for g(x) = 3x and simplify, yielding 12 - 3x.
  • **Step 4:** Finally, insert this result into f(x) = x + 1, which gives us the final result 13 - 3x.
This structured process helps digest complex mathematical tasks by focusing on one operation at a time. By following these steps, students learn both the procedure for function composition and how to seamlessly connect multiple algebraic manipulations.
Mathematical Functions
Mathematical functions are rules that assign a unique output to each input. They are fundamental in algebra and calculus.
  • In this exercise, functions f, g, and h illustrate linear transformations, each defined by its explicit rules.
  • A function like h(x) = 4 - x takes any x value and computes a corresponding result by performing arithmetic operations.
Functions are the building blocks that model relationships between variables. They can represent real-world phenomena or abstract mathematical concepts. With function composition, we combine multiple functions into one, providing a comprehensive operation that integrates each function's influence in sequence to transform the initial input stepwise. This highlights how different transformations coalesce, offering insights into how variables interrelate in more complex scenarios.

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Most popular questions from this chapter

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=1+\frac{1}{x^{2}}, \quad \text { stretched horizontally by a factor of } 3 $$

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=\sqrt{x+1}, \quad g(x)=\frac{1}{x+4}, \quad h(x)=\frac{1}{x} $$

Graph the functions in Exercises \(37-56\) $$ y=\frac{1}{x^{2}}+1 $$

Graph \(y=\sin x\) and \(y=\lceil\sin x\rceil\) together. What are the domain and range of \(\lceil\sin x\rceil ?\)

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=\sqrt{4-x^{2}}, \quad \text { stretched horizontally by a factor of } 2 $$
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