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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1:45 minutes

Problem 16j

Textbook Question

Use the table to evaluate the given compositions. <IMAGE>

ƒ(ƒ(h(3)))

Verified step by step guidance

Identify the innermost function in the composition, which is \( h(3) \).

Use the table to find the value of \( h(3) \).

Substitute the value of \( h(3) \) into the next function in the composition, \( f(h(3)) \).

Use the table to find the value of \( f(h(3)) \).

Substitute the value of \( f(h(3)) \) into the outermost function, \( f(f(h(3))) \), and use the table to find the final value.

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Key Concepts

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

Function Composition

Function composition involves applying one function to the result of another function. If you have two functions, f(x) and g(x), the composition f(g(x)) means you first apply g to x, and then apply f to the result. This concept is fundamental in calculus as it allows for the evaluation of complex expressions by breaking them down into simpler parts.

Evaluating Functions

Evaluating a function means substituting a specific input value into the function to find the output. For example, if f(x) = x + 2, then f(3) = 3 + 2 = 5. In the context of function composition, you will need to evaluate the inner function first before using its output as the input for the outer function.

Nested Functions

Nested functions occur when a function is applied within another function. In the expression f(f(h(3))), h(3) is evaluated first, then the result is used as the input for f, and finally, that output is again used as the input for f. Understanding how to handle nested functions is crucial for correctly evaluating compositions in calculus.