Algebra
Textbooks
Boundless Algebra
Exponents, Logarithms, and Inverse Functions
Inverse and Composite Functions
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions Inverse and Composite Functions
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 10
Created by Boundless

Inverses of Composite Functions

A composite function represents, in one function, the results of an entire chain of dependent functions.

Learning Objective

  • Solve for the inverse of a composite function


Key Points

    • The composition of functions is always associative. That is, if $f$, $g$, and $h$ are three functions with suitably chosen domains and co-domains, then $ f ∘ (g ∘ h) = (f ∘ g) ∘ h$, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions.
    • Functions can be inverted and then composed, giving the notation of: $(f \prime \circ g \prime ) (x)$.
    • Functions can be composed and then inverted, yielding the following notation: $(f\circ g)\prime (x)$.

Term

  • composite function

    A function of one or more independent variables, at least one of which is itself a function of one or more other independent variables; a function of a function


Full Text

Composition and Composite Functions

In mathematics, function composition is the application of one function to the results of another.

Composition of functions

g ∘ f, the composition of f and g. For example, (g ∘ f)(c) = #.

The functions $g$ and $f$ are said to commute with each other if $g ∘ f = f ∘ g$. In general, the composition of functions will not be commutative. 

A composite function represents in one function the results of an entire chain of dependent functions. For example, if a school becomes larger, the supply of food in the cafeteria must become larger. This would entail ordering more sandwiches, which means ordering more ingredients, drinks, plates, etc. The entire chain of dependent functions are the ingredients, drinks, plates, etc., and the one composite function would be putting the entire chain together in order to calculate a larger population at the school.

Inverses and Composition

If $f$ is an invertible function with domain $X$ and range $Y$, then

$f^{ -1 }(f\left( x \right) )=x$ for every $x \quad \epsilon \quad X$

This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if $Y$ coincides with the co-domain of $f$.

Example 1

Let's go through the relationship between inverses and composition in this example. let's take two functions, compose and invert them. 

Find: $(f∘g)′(x)(f\circ g)\prime (x)$

$\begin {aligned}f(x)&={x}^{2} \\g(x)&=x+1 \end {aligned}$

$$First, compose them:

$\begin {aligned}(f\circ g)(x)&=f(x+1)=(x+1)^2 \\y&=(x+1)^2 \\\sqrt {y}&=x+1 \\\sqrt{y}-1&=x \end {aligned}$

$$Then, invert it:

$\begin {aligned} \sqrt{x}-1&=y \\\sqrt{x}-1&=(f\circ g)\prime (x) \end {aligned}$

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