Textbook Question
Composite functions and notation
Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
F(g(y))
0. Functions
Combining Functions
1:47 minutesProblem 1.43
Textbook Question
Working with composite functions
Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.
h(x) = (x³ - 5)¹⁰
Verified step by step guidance1
Step 1: Identify the structure of the composite function. The given function is \( h(x) = (x^3 - 5)^{10} \). This suggests that there is an inner function and an outer function involved.
Step 2: Determine the inner function \( g(x) \). Look for a function inside another function. Here, \( g(x) = x^3 - 5 \) is a natural choice for the inner function because it is the expression inside the power.
Step 3: Determine the outer function \( f(u) \). The outer function is applied to the result of the inner function. In this case, \( f(u) = u^{10} \) is the outer function, where \( u = g(x) = x^3 - 5 \).
Step 4: Verify the composition. Check that \( f(g(x)) = f(x^3 - 5) = (x^3 - 5)^{10} \) matches the original function \( h(x) \).
Step 5: Conclude the choices. The functions \( f(u) = u^{10} \) and \( g(x) = x^3 - 5 \) are valid choices for the outer and inner functions, respectively, such that \( h(x) = f(g(x)) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composite Functions
A composite function is formed when one function is applied to the result of another function. It is denoted as (ƒ o g)(x) = ƒ(g(x)), where g is the inner function and ƒ is the outer function. Understanding how to decompose a function into its components is essential for solving problems involving composite functions.
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Function Notation
Function notation is a way to represent functions and their relationships. In this context, h(x) represents the output of the function h for a given input x. Recognizing how to manipulate and interpret function notation is crucial for identifying the inner and outer functions in a composite function.
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Power Functions
Power functions are functions of the form f(x) = x^n, where n is a real number. In the given function h(x) = (x³ - 5)¹⁰, the outer function can be identified as a power function, while the inner function can be the expression inside the parentheses. Understanding power functions helps in determining how to break down the composite function into its components.
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