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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2:10 minutes

Problem 39d

Textbook Question

Composition of Functions

In Exercises 39 and 40, find

d. (g ○ g) (x).

ƒ(x) = 1/x , g(x) = 1/√ x + 2

Verified step by step guidance

First, understand the composition of functions. The notation (g ○ g)(x) means you need to apply the function g to itself, i.e., g(g(x)).

Start by substituting g(x) into itself. Since g(x) = 1/√(x + 2), replace x in g(x) with g(x) itself.

This substitution gives you g(g(x)) = 1/√(g(x) + 2). Now, substitute g(x) = 1/√(x + 2) into this expression.

You will have g(g(x)) = 1/√((1/√(x + 2)) + 2). Simplify the expression inside the square root.

Finally, simplify the entire expression to find the composition (g ○ g)(x). Remember to handle the square roots and fractions carefully during simplification.

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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 combining two functions to create a new function. The notation (f ○ g)(x) means applying function g to x first, and then applying function f to the result of g(x). This concept is essential for understanding how to evaluate expressions like (g ○ g)(x), where the function g is applied to itself.

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 composing functions, it is crucial to consider the domain of the inner function, as it can affect the overall composition. For example, in g(x) = 1/√(x + 2), the input x must be greater than or equal to -2 to avoid taking the square root of a negative number.

Evaluating Functions

Evaluating functions involves substituting a specific value into the function's formula to find the corresponding output. In the context of the question, evaluating (g ○ g)(x) requires first calculating g(x) and then substituting that result back into g. This step-by-step evaluation is fundamental for accurately determining the final output of the composed function.

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Related Practice

Textbook Question

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

Textbook Question

Inverse of composite functions

c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.

Textbook Question

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).

b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?

Textbook Question

Composition of Functions

In Exercises 39 and 40, find

a. (ƒ ○ g) (-1).

ƒ(x) = 1/x , g(x) = 1/√ x + 2

Textbook Question

In Exercises 41 and 42, (a) write formulas for ƒ ○ g and g ○ ƒ and find the (b) domain and (c) range of each.

ƒ(x) = 2 - x², g(x) = √ x + 2

Textbook Question

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)

x² |x|²

Textbook Question

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)

x³ |x³|

Textbook Question

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)

4 - x² |4 - x²|

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Composition of FunctionsIn Exercises 39 and 40, findd. (g ○ g) (x).ƒ(x) = 1/x , g(x) = 1/√ x + 2

Key Concepts

Function Composition

Domain of a Function

Evaluating Functions

Watch next

Composition of Functions

In Exercises 39 and 40, find

d. (g ○ g) (x).

ƒ(x) = 1/x , g(x) = 1/√ x + 2