Composite Functions
Introduction
After learning about the definition of a function we learned about how to evaluate a function at a real number. Recalling how this is defined, if we want to evaluate a function f(x) at x = 5, we would replace all occurrences of x with 5, and simplify. For composite functions, instead of replacing the independent variable, usually x, with a number, we replace it with a function.
Definition and notation
Given two functions, f and g, the composite function, denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\circ g} , is a function where .
Example:
Suppose Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x)={\frac {1+x}{x-3}}{\text{ and }}g(x)={\sqrt {x}}<math>Then<math>()f\circ g)(x)={\frac {1+{\sqrt {x}}}{{\sqrt {x}}-3}}}
Domain
The domain of a composite function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (f\circ g)(x)} is the collection of x-values in the domain of g such that g(x) is in the domain of f.
Example: Find the domain of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\circ g{\text{ if }}f(x)={\frac {1}{x+1}}{\text{ and }}g(x)={\frac {1}{x+3}}}
We start by noting that the domain of g(x) is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-\infty ,3)\cup (3,\infty )} . Now we want to know for what values of x is g(x) = -1. So we solve: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 = \frac{1}{x + 3}} . Solving this equation we find that g(-4) = -1. So -4 must be removed from the domain of g to result in th domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ g} . To finish the problem: the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\circ g \text{ is }(-\infty, -4)\cup (-4, 3) \cup (3, \infty)}