1. Operations on Functions:
   • Addition/Subtraction: Given two functions
     $$f(x)
     

     g(x), their sum or difference is a new function 

     
     $$

     h(x)=f(x)±g(x), where
     $ℎ<br>$

     h(x) is defined as 

     h(x)=f(x)±g(x) for all 

     
     $$

     x in the domain where both
     $\mathrm{<br>}$

     f(x) and g(x) are defined.

   • Multiplication/Division: The product or quotient of two functions

     $$

     f(x) and
     $$

     g(x) is a new function
     $\mathrm{<br>}$

     h(x)=f(x)⋅g(x) or ℎ

     
     $h(x)=g(x)f(x)​, respectively, where$

   • 
     $ℎ$

     h(x) is defined as 
     $$h(x)=f(x)⋅g(x) or h(x)=g(x)f(x)​
     

     x in the domain where both 

     f(x) and 

     g(x) are defined.

   • Composition: Given two functions
     

     $f(x)$ and $g(x)$, their composition is a new function 

     
     $ℎ($

     $h(x)=f(g(x))$, where
     $$

     $h(x)$ is defined as
     $$

     $h(x)=f(g(x))$ for all
     $$

     $x$ in the domain where
     $$

     $g(x)$ is defined, and
     $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">g</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">x</span><span\; class="mclose">)</span></span></span></span>is\; in\; the\; domain\; of </span>}$

     
     $$

     $f(x)$.

   • Inverse: For a function
     

     $f(x)$, its inverse function 

     
     ${}^{}$

     f−1(x) is a function that “undoes” the action of
     $$

     $f(x)$, such that
     $$

     f(f−1(x))=x for all
     $$

     $x$ in the domain of
     ${}^{}$

     f−1(x).

2. Recursively Defined Functions:
   • A recursively defined function is a function that is defined in terms of itself. The definition of the function contains one or more base cases and one or more recursive cases.
   • Base Case: The base case(s) provide the initial condition(s) or values for the function. It’s the stopping criterion for the recursion.
   • Recursive Case: The recursive case(s) define the function in terms of itself, usually involving smaller or simpler instances of the problem.
   • Recursive functions are often used to define sequences, series, and other mathematical constructs. Examples include the Fibonacci sequence, factorial function, and Ackermann function.

Understanding these concepts is essential for working with functions in various mathematical contexts, including calculus, discrete mathematics, and computer science.