Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Combining Functions

Calculus

0. Functions

Combining Functions

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2:25 minutes

Problem 19d

Textbook Question

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find the domain of g o ƒ.

Verified step by step guidance

The composition of functions \( g \circ f \) means applying function \( g \) to the result of function \( f \). In this case, \( g \circ f(x) = g(f(x)) = g(x^3) = \sin(x^3).\)

The function \( f(x) = x^3 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \).

The sine function, \( \sin x \), is defined for all real numbers. Therefore, the domain of \( g(x) \) is also all real numbers, \( (-\infty, \infty) \).

Since \( f(x) = x^3 \) is defined for all real numbers and \( g(x) = \sin x \) is also defined for all real numbers, the composition \( g \circ f(x) = \sin(x^3) \) is defined for all real numbers.

The domain of \( g \circ f \) is the set of all real numbers, \( (-\infty, \infty) \), because both \( f(x) \) and \( g(x) \) are defined for all real numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, g o ƒ means we first apply the function ƒ to x, and then apply g to the result of ƒ(x). Understanding how to combine functions is essential for determining the overall behavior and properties of the composite function.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For composite functions, the domain is influenced by both the inner function and the outer function. To find the domain of g o ƒ, we must ensure that the output of ƒ(x) falls within the domain of g(x).

Function Behavior and Restrictions

Different functions have specific restrictions that affect their domains. For example, the function g(x) = sin x is defined for all real numbers, while ƒ(x) = x³ is also defined for all real numbers. However, if we were to consider a function like h(x) = √x, it would impose restrictions since it is only defined for x ≥ 0. Understanding these behaviors is crucial for determining the valid inputs for composite functions.

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Related Practice

Textbook Question

Composition of polynomials

Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.

What is the degree of the following polynomials?

ƒ o g

Textbook Question

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Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Evaluate h(g( π/2)).

Textbook Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find h (ƒ (x)).

Textbook Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find ƒ(g(h( x))).

Open Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find the domain of ƒ o g.

Textbook Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find the domain of ƒ o g.

Textbook Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

Textbook Question

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = x⁴ - 12x² + 30

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Composite functionsLet ƒ(x) = x³, g (x) = sin x and h(x) = √x.Find the domain of g o ƒ.

Key Concepts

Composite Functions

Domain of a Function

Function Behavior and Restrictions

Watch next

Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find the domain of g o ƒ.