2.3 Composition and Inverse Functions

Function composition and inverses are like mathematical magic tricks. You combine functions to create new ones or find a function that undoes another. It's all about understanding how functions relate and interact with each other.

These concepts are crucial for analyzing complex relationships in real-world scenarios. They help us model and solve problems in various fields, from physics to economics, by breaking down complicated systems into simpler, manageable parts.

Function Composition

Combining Functions

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• Function composition combines two or more functions to create a new function
  • The output of one function becomes the input of the next function
• The composition of functions f and g is denoted as $(f ∘ g)(x) = f(g(x))$
  • The function g is applied first, then the function f is applied to the result
• To find the composite function $(f ∘ g)(x)$, substitute the function $g(x)$ for x in the function $f(x)$

Domain and Range Considerations

• The domain of the composite function $(f ∘ g)(x)$ is the set of all x-values for which both $g(x)$ and $f(g(x))$ are defined
• When composing functions, the range of the inner function (g) must be a subset of the domain of the outer function (f) for the composition to be defined
• The order of composition matters; in general, $(f ∘ g)(x) ≠ (g ∘ f)(x)$, unless f and g are inverse functions

Inverse Function Existence

One-to-One Functions

• An inverse function is a function that "undoes" the original function, denoted as $f⁻¹(x)$
• For a function f to have an inverse, it must be a one-to-one (injective) function
  • Each element in the codomain is paired with at most one element in the domain
• A function is one-to-one if it passes the horizontal line test
  • Any horizontal line intersects the graph of the function at most once

Restricting Domain for Inverses

• If a function is not one-to-one, it may be possible to restrict its domain to create a one-to-one function that has an inverse
• The graph of a function and its inverse are symmetric about the line $y = x$

Finding Inverses

Algebraic Method

• To find the inverse of a function algebraically:
  1. Replace $f(x)$ with y
  2. Swap x and y
  3. Solve the equation for y
  4. Replace y with $f⁻¹(x)$
• The domain of the inverse function is the range of the original function
• The range of the inverse function is the domain of the original function

Graphical Method

• To find the inverse of a function graphically:
  1. Reflect the graph of the original function across the line $y = x$
  2. If the original function is not one-to-one, restrict its domain to obtain a one-to-one function before reflecting
• The inverse of a linear function with a slope m and y-intercept b is another linear function
  • The inverse has a slope $1/m$ and y-intercept $-b/m$

Applications of Composition and Inverses

Properties and Equations

• The composition of a function with its inverse results in the identity function: $(f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x$
• The inverse of the inverse of a function is the original function: $(f⁻¹)⁻¹ = f$
• The inverse of a composition of functions is the composition of their inverses in the reverse order: $(f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹$
• Composition and inverses can be used to solve equations involving functions
  • Examples: $f(g(x)) = c$ or $f(x) = g(x)$

Real-World Examples

• Converting between different units of measurement (Celsius to Fahrenheit, miles to kilometers)
• Modeling relationships between variables in various fields
  • Physics (velocity and acceleration, force and displacement)
  • Economics (supply and demand, cost and revenue)
  • Social sciences (population growth, voting patterns)
• Analyzing the behavior of systems and processes that involve multiple stages or transformations (manufacturing processes, chemical reactions)