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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2:30 minutes

Problem 39

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.
F(F(x))

Verified step by step guidance

Identify the function F(x) which is given as F(x) = \frac{1}{x-3}.

To find F(F(x)), substitute F(x) into itself. This means replacing x in F(x) with F(x).

Substitute F(x) = \frac{1}{x-3} into F(x) to get F(F(x)) = \frac{1}{\frac{1}{x-3} - 3}.

Simplify the expression \frac{1}{\frac{1}{x-3} - 3} by finding a common denominator for the terms in the denominator.

The common denominator is x-3, so rewrite the expression as \frac{1}{\frac{1 - 3(x-3)}{x-3}} and simplify further.

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Key Concepts

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

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In this case, F(F(x)) means that we first evaluate F at x, and then take that result and use it as the input for F again. Understanding how to combine functions is crucial for simplifying expressions involving multiple functions.

Function Notation

Function notation is a way to denote functions and their operations clearly. For example, F(x) represents the function F evaluated at x. Recognizing how to read and interpret function notation is essential for working with composite functions and understanding the relationships between different functions.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with. This can include combining like terms, factoring, or substituting values. In the context of composite functions, simplifying F(F(x)) requires substituting the expression for F(x) into itself and then simplifying the resulting expression.