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Chapter 3: Problem 39

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=\sqrt[3]{x}, \quad g(x)=\sqrt[4]{x} $$

Short Answer

Expert verified
All compositions are defined for non-negative \(x\) except \(f \circ f\), which is defined for all real numbers.

Step by step solution

01

Understand function composition

Function composition involves applying one function to the results of another. Specifically, for two functions \(f\) and \(g\), the compositions \((f\circ g)(x)\) and \((g\circ f)(x)\) represent \(f(g(x))\) and \(g(f(x))\) respectively.
02

Compute \(f \circ g(x)\)

Start by substituting \(g(x)=\sqrt[4]{x}\) into \(f\). This gives:\[(f\circ g)(x) = f(g(x)) = \sqrt[3]{g(x)} = \sqrt[3]{\sqrt[4]{x}} = x^{1/12}\].
03

Find the domain of \(f \circ g\)

The composition \(f \circ g = x^{1/12}\) involves roots. Both cube and fourth roots accept non-negative numbers, so the domain is all non-negative \(x\), which is \(x \geq 0\).
04

Compute \(g \circ f(x)\)

Substitute \(f(x)=\sqrt[3]{x}\) into \(g\). This gives:\[(g\circ f)(x) = g(f(x)) = \sqrt[4]{f(x)} = \sqrt[4]{\sqrt[3]{x}} = x^{1/12}\].
05

Find the domain of \(g \circ f\)

Similarly to \(f \circ g\), since the expression \(x^{1/12}\) involves both cube and fourth roots, its domain is all non-negative \(x\), which is again \(x \geq 0\).
06

Compute \(f \circ f(x)\)

Substitute \(f(x)=\sqrt[3]{x}\) into itself. This gives:\[(f\circ f)(x) = f(f(x)) = \sqrt[3]{\sqrt[3]{x}} = x^{1/9}\].
07

Find the domain of \(f \circ f\)

The function \(x^{1/9}\) is defined for all real \(x\) because cube roots can take any real number. Thus, the domain is all real numbers \((-\infty, \infty)\).
08

Compute \(g \circ g(x)\)

Substitute \(g(x)=\sqrt[4]{x}\) into itself. This gives:\[(g\circ g)(x) = g(g(x)) = \sqrt[4]{\sqrt[4]{x}} = x^{1/16}\].
09

Find the domain of \(g \circ g\)

The expression \(x^{1/16}\) is defined for non-negative \(x\) because it involves the fourth root. Thus, the domain is \(x \geq 0\).

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Key Concepts

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

Cube Roots
A cube root is a number that, when multiplied by itself three times, gives the original number. For any real number \( x \), its cube root is denoted by \( \sqrt[3]{x} \). Unlike square roots, cube roots can be evaluated for negative numbers. This is because the product of three negative numbers is also negative. Hence, the cube root of \(-8\) is \(-2\), as \(-2 \times -2 \times -2 = -8\).

Understanding the nature of cube roots is essential, especially when dealing with functions. In the case of function composition, the domain of \( f(x) = \sqrt[3]{x} \) includes all real numbers. This means you can input any number into this function without restriction. This feature makes cube roots quite versatile in mathematical operations, like when you compose functions.
Fourth Roots
A fourth root is a number that, when multiplied by itself four times, results in the original number. It is represented as \( \sqrt[4]{x} \). However, unlike cube roots, fourth roots are restricted to non-negative numbers only.

This limitation occurs because the product of a negative number repeated an even number of times is positive, and a negative input for a fourth root does not produce a real number. Therefore, the domain of a fourth root function \( g(x) = \sqrt[4]{x} \) is limited to \( x \geq 0 \).

Fourth roots often appear in mathematics when simplifying expressions or solving equations. Their properties are crucial when working on function compositions, especially when finding the domain of more complex expressions derived from these roots.
Domain of Functions
The domain of a function refers to all the possible input values (\( x \)) for which the function is defined. Each function type has its domain rules based on its mathematical properties. For example, square and fourth root functions can only accept non-negative numbers. However, cube root functions can accept any real number because they accommodate negative inputs as well.

When you compose functions, determining the domain of the resulting function involves understanding the domain restrictions of each individual function. Let's take \( f(x) = \sqrt[3]{x} \) and \( g(x) = \sqrt[4]{x} \) as examples. The function \( (f \circ g)(x) \) becomes \( f(g(x)) \). Here, \( g(x) = \sqrt[4]{x} \) limits the inputs to \( x \geq 0 \), which carries over to the composed function. Similarly, when you find \( g \circ f(x) \), you consider the domain of \( g \), keeping \( x \geq 0 \) as the restriction even after composition.

Ultimately, understanding domains in function compositions helps streamline solving and simplifies error identification in mathematical analysis.

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Most popular questions from this chapter

61鈥�68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = x^2$$

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