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(F(x))
0. Functions
Combining Functions
1:39 minutesProblem 1.1.45
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) = √ (x⁴ + 2 )
Verified step by step guidance1
Step 1: Understand the problem. We need to express the function \( h(x) = \sqrt{x^4 + 2} \) as a composition of two functions \( f \) and \( g \), such that \( h(x) = (f \circ g)(x) = f(g(x)) \).
Step 2: Identify the inner function \( g(x) \). Look for a part of \( h(x) \) that can be isolated as a simpler function. In this case, consider \( g(x) = x^4 + 2 \).
Step 3: Determine the outer function \( f(x) \). Since \( g(x) = x^4 + 2 \), the remaining operation in \( h(x) \) is taking the square root. Therefore, let \( f(x) = \sqrt{x} \).
Step 4: Verify the composition. Substitute \( g(x) \) into \( f(x) \) to ensure it reconstructs \( h(x) \): \( f(g(x)) = f(x^4 + 2) = \sqrt{x^4 + 2} \), which matches \( h(x) \).
Step 5: Conclude that the functions \( f(x) = \sqrt{x} \) and \( g(x) = x^4 + 2 \) 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 (f o g)(x) = f(g(x)), where f is the outer function and g is the inner function. Understanding how to decompose a function into its components is essential for identifying suitable outer and inner functions that yield the desired composite function.
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Function Decomposition
Function decomposition involves breaking down a complex function into simpler constituent functions. This process is crucial when working with composite functions, as it allows us to identify potential candidates for the inner and outer functions. For example, in the function h(x) = √(x⁴ + 2), recognizing the structure of the expression can guide us in selecting appropriate f and g.
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Square Root Function
The square root function, denoted as √x, is a fundamental mathematical function that returns the non-negative value whose square equals x. In the context of composite functions, it often serves as an outer function. Understanding its properties, such as its domain and range, is vital for determining how it can be combined with other functions to form a composite function like h(x).
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