Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 1.R.17

Chapter 1, Problem 1.R.17

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))

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: Use the property of odd functions to find f(-3). Since f is odd, f(-3) = -f(3).

Step 3: Calculate 1 + f(-3) using the result from Step 2. This becomes 1 - f(3).

Step 4: Understand the property of one-to-one functions. A function is one-to-one if it has an inverse, meaning each output is mapped from a unique input.

Step 5: Use the inverse function property to find f^{-1}(1 + f(-3)). Since 1 + f(-3) = 1 - f(3), find the x such that f(x) = 1 - f(3).

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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 means that if you know the value of the function at a positive input, you can easily determine its value at the corresponding negative input. Understanding this property is crucial for evaluating expressions involving odd functions, such as f(-3) in the given question.

Inverse Functions

An inverse function, denoted as f-1, reverses the effect of the original function. If f(x) = y, then f-1(y) = x. For one-to-one functions, each output corresponds to exactly one input, allowing for the existence of an inverse. This concept is essential for solving the expression f-1(1 + f(-3)), as it requires finding the input that produces a specific output.

One-to-One Functions

A one-to-one function is a function where each output is produced by exactly one input, meaning f(a) = f(b) implies a = b. This property ensures that the function has an inverse. In the context of the question, knowing that both f and g are one-to-one allows us to confidently use their inverses without ambiguity, which is critical for evaluating the function values requested.

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One-Sided Limits

Related Practice

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

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

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = 7 / (x + 3)

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Textbook Question

Solving equations Solve each equation.

ln 3x + ln (x + 2) = 0

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

If y= 3ˣ , then x = ³√y

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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

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))

Verified video answer for a similar problem:

Key Concepts

Odd Functions

Inverse Functions

One-to-One Functions

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))