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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.96c
Chapter 1, Problem 1.96c

Inverse of composite functions


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

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To determine if the inverse of a composite function \( f(x) = g(h(x)) \) exists, we first need to understand the concept of one-to-one functions. A function is one-to-one (injective) if each output is mapped from a unique input, meaning no two different inputs produce the same output.
Given that both \( g \) and \( h \) are one-to-one functions, we can infer that \( h(x) \) maps each input \( x \) to a unique output \( y \), and then \( g(y) \) maps each \( y \) to a unique output \( z \). This ensures that the entire mapping from \( x \) to \( z \) through \( g(h(x)) \) is also one-to-one.
Since \( f(x) = g(h(x)) \) is a composition of two one-to-one functions, it is itself one-to-one. This is because the composition of two injective functions is injective.
For a function to have an inverse, it must be both one-to-one and onto (bijective). In this context, we are primarily concerned with the injective property, which is satisfied as shown.
Therefore, since \( f(x) = g(h(x)) \) is one-to-one, it has an inverse function \( f^{-1}(x) \), which can be found by reversing the operations of \( g \) and \( h \) in the reverse order, i.e., \( f^{-1}(x) = h^{-1}(g^{-1}(x)) \).

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Key Concepts

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

One-to-One Functions

A function is considered one-to-one (injective) if it assigns distinct outputs to distinct inputs. This means that for any two different inputs, the outputs will also be different. One-to-one functions are crucial for the existence of inverses because they ensure that each output corresponds to exactly one input, preventing ambiguity in reversing the function.
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Composite Functions

A composite function is formed when one function is applied to the result of another function, denoted as ƒ(x) = g(h(x)). The inner function h(x) is evaluated first, followed by the outer function g. Understanding composite functions is essential for analyzing the behavior of combined transformations and determining the conditions under which their inverses exist.
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Existence of Inverses

For a function to have an inverse, it must be bijective, meaning it is both one-to-one and onto. In the case of composite functions, if both g and h are one-to-one, then their composition g(h(x)) will also be one-to-one. This guarantees that the inverse function exists, as each output from the composite function can be traced back to a unique input.
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Related Practice
Textbook Question

A culture of bacteria has a population of 150150 cells when it is first observed. The population doubles every 12 hr12~\(\text{hr}\), which means its population is governed by the function p(t)=150⋅2t12p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}, where tt is the number of hours after the first observation.

What is the population 4 days4~\(\text{days}\) after the first observation?

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Textbook Question

Identify the symmetry (if any) in the graphs of the following equations.

y2−4x2=4y^2-4x^2=4

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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

A culture of bacteria has a population of 150150 cells when it is first observed. The population doubles every 12 hr12~\(\text{hr}\), which means its population is governed by the function p(t)=150⋅2t12p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}, where tt is the number of hours after the first observation.

How long does it take the population to triple in size?

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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)

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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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