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 12

Chapter 1, Problem 12

Find functions ƒand g such that ƒ(g(x)) = (x² +1)⁵ . Find a different pair of functions ƒ and g that also satisfy ƒ(g(x)) = (x² +1)⁵

Verified step by step guidance

Consider the composition of functions $ f(g(x)) = (x^2 + 1)^5 $. We need to find functions $ f $ and $ g $ such that their composition results in this expression.

First, let's choose $ g(x) = x^2 + 1 $. This simplifies the inner function.

Now, we need to find $ f $ such that $ f(g(x)) = f(x^2 + 1) = (x^2 + 1)^5 $. This implies $ f(u) = u^5 $ where $ u = g(x) $.

For a different pair, let's choose $ g(x) = x $. This means we need $ f(x) = (x^2 + 1)^5 $ to satisfy the composition.

Thus, the first pair is $ f(u) = u^5 $ and $ g(x) = x^2 + 1 $, and the second pair is $ f(x) = (x^2 + 1)^5 $ and $ g(x) = x $.

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Key Concepts

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

Composition of Functions

The composition of functions involves combining two functions, where the output of one function becomes the input of another. This is denoted as ƒ(g(x)), meaning you first apply g to x, and then apply ƒ to the result. Understanding this concept is crucial for solving the problem, as it requires finding two functions that, when composed, yield a specific result.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function (x² + 1)⁵ is a polynomial function raised to a power, which can be expanded using the binomial theorem. Recognizing the structure of polynomial functions helps in identifying potential forms for ƒ and g that can achieve the desired composition.

Function Decomposition

Function decomposition is the process of breaking down a complex function into simpler component functions. In this context, it involves finding two distinct functions ƒ and g such that their composition results in (x² + 1)⁵. This concept is essential for exploring different pairs of functions that can satisfy the given equation, allowing for creative solutions and insights into function behavior.

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Properties of Functions

Related Practice

Textbook Question

If ƒ(x) = √x and g(x) = x³-2 and , simplify the expressions (ƒ o g) (3), (ƒ o ƒ) (64), (g o ƒ) (x) and (ƒ o g) (x)

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Textbook Question

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>

Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.

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Textbook Question

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

ƒ(g(4))

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Textbook Question

Use the graph of $f$ in the figure to plot the following functions.

<IMAGE>

$y=f$\left$(x-2$\right$)$

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Textbook Question

Solve the equation sin 2Θ = 1, for 0 ≤ Θ < 2π .

Textbook Question

How do you obtain the graph of $y=f$\left$(3x$\right$)$ from the graph of $y=f$\left$(x$\right$)$ ?