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Chapter 2: Problem 4

Let \(f(x)=x^{3}+5, g(x)=x^{2}-2\), and \(h(x)=2 x+4 .\) Find the rule for each function. \(g f\)

Short Answer

Expert verified
The composite function \(g(f(x))\) is \(g(f(x)) = (x^3 + 5)^2 - 2\).

Step by step solution

01

Rewrite given functions

We have the following functions: \(f(x) = x^3 + 5\) \(g(x) = x^2 - 2\) We want to find the composite function, \(g(f(x))\). 2. Get the result of \(f(x)\)
02

Get the result of \(f(x)\)

Firstly, we need to find the result of \(f(x)\). We have: \(f(x) = x^3 + 5\) 3. Plug \(f(x)\) as input of \(g(x)\)
03

Plug \(f(x)\) as input of \(g(x)\)

Now we will plug the output of \(f(x)\) into the input of \(g(x)\): \(g(f(x)) = g(x^3 + 5)\) 4. Calculate \(g(f(x))\)
04

Calculate \(g(f(x))\)

Now we need to replace the input in \(g(x)\) function with the result of \(f(x)\): \(g(f(x)) = (x^3 + 5)^2 - 2\) Hence, the composite function \(g(f(x))\) is: \[g(f(x)) = (x^3 + 5)^2 - 2\]

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Key Concepts

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

Function Composition
Function composition is essentially taking the output of one function and using it as the input for another function.
This is often denoted as \(g(f(x))\), where \(f(x)\) is the inner function, and \(g(x)\) is the outer function.
In this exercise, we use the result from the function \(f(x) = x^{3}+5\), and insert it into the function \(g(x) = x^{2}-2\).
  • First, solve \(f(x)\) to get \(x^{3} + 5\).
  • Next, replace 鈥渪鈥� in \(g(x)\) with the result, \(x^{3} + 5\).
By doing this, you create a new function, \(g(f(x))\), which helps in seeing the transformation from one function to another process.
This approach is powerful for analyzing combined processes or effects in mathematics and real-world applications.
Mathematical Functions
A mathematical function is a unique relationship between a set of inputs and outputs.
Each input is associated with exactly one output; think of it as a mathematical machine.
For example, in this scenario, we have the functions:
  • \(f(x) = x^{3}+5\)
  • \(g(x) = x^{2}-2\)
  • \(h(x) = 2x+4\)
Each function has its rule, deciding how the input is transformed to an output.
Understanding these simple rules and their effects helps interpret and apply mathematical concepts effectively.
This structured input and output methodology is fundamental not only in pure mathematics but also in practical fields like engineering and computer science.
Algebraic Functions
Algebraic functions are expressions composed using algebraic operations like addition, subtraction, multiplication, division, and exponentiation to defined exponents.
In this exercise, both \(f(x)\) and \(g(x)\) are examples of algebraic functions.
  • \(f(x) = x^{3} + 5\) involves both exponentiation and addition.
  • \(g(x) = x^{2} - 2\) involves exponentiation and subtraction.
These serve as the building blocks for constructing more complex expressions, like the composite function \(g(f(x))\).
Algebraic functions form the basis of many calculus concepts and are essential in modeling real-world phenomena where relationships between variables must be defined and analyzed.
Recognizing how these functions operate individually and together is key to mastering more advanced subjects in mathematics.

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Most popular questions from this chapter

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