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

Common Functions

Calculus

0. Functions

Common Functions

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2:28 minutes

Problem 1.R.12

Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>
f⁻¹ (10)

Verified step by step guidance

Step 1: Understand the properties of odd functions. An odd function satisfies the condition f(-x) = -f(x) for all x in the domain of f.

Step 2: Recognize that f is one-to-one, meaning that each y-value in the range of f corresponds to exactly one x-value in the domain of f.

Step 3: Recall that the inverse function f⁻¹(y) gives the x-value such that f(x) = y.

Step 4: Since f is odd, if f(a) = 10, then f(-a) = -10. Use this property to find the x-value that corresponds to f(x) = 10.

Step 5: Use the graph of f to locate the x-value where f(x) = 10. This x-value is f⁻¹(10).

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Key Concepts

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

Odd Functions

An odd function is defined by the property f(-x) = -f(x) for all x in its domain. This symmetry about the origin implies that the graph of an odd function will reflect across both axes. Understanding this property is crucial for analyzing the behavior of the function and its inverse, especially when determining specific function values.

One-to-One Functions

A one-to-one function, or injective function, is one where each output is produced by exactly one input. This means that if f(a) = f(b), then a must equal b. This property is essential for the existence of an inverse function, as it ensures that the inverse will also be a function, allowing us to find values like f⁻¹(10) without ambiguity.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If f(x) gives an output y, then f⁻¹(y) will return the input x. For one-to-one functions, the inverse can be found by swapping the roles of x and y in the equation y = f(x). Understanding how to find and interpret inverse functions is key to solving problems involving function values like f⁻¹(10).

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Related Practice

Textbook Question

Inverse of composite functions

a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure) <IMAGE>.

a. Find the domain and a formula for each function.

Textbook Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure)<IMAGE>.

b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)

Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f⁻¹( g⁻¹(4))

Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f^-1(1 + f(-3))

Textbook Question

A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs $5/ft for the legs and $10/ft for the hypotenuse, write the total cost C of construction as a function of h.

Textbook Question

Theory and Examples

The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

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