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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1:58 minutes

Problem 15e

Textbook Question

Use the graphs of ƒ and g in the figure to determine the following function values. y = f(x) ; y=g(x) <IMAGE>

e. ƒ(ƒ(8))

Verified step by step guidance

Step 1: Identify the value of f(8) using the graph of the function f(x). Locate the point on the graph where x = 8 and find the corresponding y-value, which is f(8).

Step 2: Once you have determined f(8), use this value as the new input for the function f(x). This means you need to find f(f(8)).

Step 3: Locate the value of f(f(8)) on the graph. Use the y-value obtained from Step 1 as the x-coordinate on the graph of f(x) and find the corresponding y-value.

Step 4: Verify your steps by checking the graph to ensure that the values you have used and found are consistent with the graph's data.

Step 5: Conclude by stating the value of f(f(8)) based on the graph analysis.

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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. In this case, ƒ(ƒ(8)) means you first find the value of ƒ at 8, and then use that result as the input for ƒ again. Understanding how to compose functions is crucial for solving problems that require multiple evaluations.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. For example, to evaluate ƒ(8), you look at the graph of ƒ and find the corresponding y-value when x equals 8. This concept is fundamental in determining specific function values from their graphs.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of functions to extract information about their behavior and values. In this question, understanding how to read the graphs of ƒ and g is essential for accurately determining the function values needed for the composition. This skill is vital for connecting algebraic expressions with their graphical counterparts.