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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2:06 minutes

Problem 47

Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o g

Verified step by step guidance

Identify the composite function \( (f \circ g)(x) \), which means \( f(g(x)) \).

Substitute \( g(x) = x^2 - 4 \) into \( f(x) = |x| \), so \( f(g(x)) = |x^2 - 4| \).

Determine the domain of \( g(x) = x^2 - 4 \), which is all real numbers since a polynomial is defined everywhere.

Since \( f(x) = |x| \) is also defined for all real numbers, the domain of \( f(g(x)) = |x^2 - 4| \) is all real numbers.

Conclude that the composite function \( (f \circ g)(x) = |x^2 - 4| \) is 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. It is denoted as (f o g)(x) = f(g(x)). Understanding how to combine functions is essential for evaluating composite functions, as it requires substituting the output of g(x) into f(x).

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. When dealing with composite functions, it is crucial to determine the domain of both the inner and outer functions, as the overall domain is influenced by any restrictions from either function.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is important in the context of composite functions because it can affect the overall behavior and domain of the composite function, particularly when combined with other functions that may have their own restrictions.

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4, g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

ƒ (√(x+4))

Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (x³ - 5)¹⁰

Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

Textbook Question

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g .

h(x) = √ (x⁴ + 2 )

Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

ƒ o G

Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o g o ƒ

Textbook Question

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.

G o G

Textbook Question

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.

(ƒ o g) (x) = x⁴ + 6x² + 9

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