Function composition is a fundamental mathematical concept where one function is applied to the results of another, denoted as h(x) = f(g(x)). This text explores how to construct and evaluate composite functions, domain considerations, and their applications in various fields such as physics and economics. It also touches on composing functions with fractions, multivariables, and the decomposition and inversion of functions.
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Function composition is a fundamental mathematical operation that involves applying one function to the results of another
Notation
Function composition is represented as \( h(x) = f(g(x)) \) or \( h(x) = (f \circ g)(x) \), signifying the application of one function to the output of another
Process
To construct a composite function, one inserts the output from the inner function into the outer function
Function composition can be illustrated through examples such as \( h(x) = 2(x + 3) = 2x + 6 \) and \( h(x) = 3(4x - 1) + 1 = 12x - 2 \)
Evaluating a composite function involves substituting a specific value for the variable and computing the outcome
Domain constraints
The domain of a composite function is limited by the domains of the individual functions it comprises
Range restrictions
The range of the inner function must be within the domain of the outer function in a composite function
When composing functions with fractions or multiple variables, it is essential to consider domain restrictions and adhere to the principle of substitution
Decomposing a function involves breaking it down into simpler constituent functions
Functions can be decomposed into simpler constituent functions, such as \( h(x) = 4x^2 + 3 \) into \( f(x) = 4x \) and \( g(x) = x^2 + \frac{3}{4} \)
Function composition has practical applications in various disciplines, such as physics and economics, for modeling and solving real-world problems
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Composite function definition
A composite function is created when one function is applied to the result of another function.
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Domain exclusion in composite functions
When composing functions, values causing undefined operations, like division by zero, are excluded from the domain.
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Composite functions with multivariables
In a multivariable composite function, the output only includes variables from the inner function.
