Theory of Computation: Question Set – 04

What is a bijective function?

A function that is both one-to-one and onto is called a bijective function. A bijective function is also called a one-to-one correspondence, or a bijection. If a function is bijective, then there is a unique inverse function that maps the elements of the range back to the elements of the domain.

What is the composition of functions?

The composition of two functions f and g is a new function that applies f to the output of g. In other words, if f(x) is the first function and g(x) is the second function, then the composition of f and g is written as f(g(x)) and is defined as f(g(x)) = f(g(x)). The domain of the composition is the set of all x in the domain of g such that g(x) is in the domain of f.

What is the inverse of a function?

The inverse of a function f is a new function that “undoes” the effect of f. In other words, if y = f(x), then the inverse function f-1(y) maps y back to x. The inverse of a function exists if and only if the function is bijective. If f is bijective, then f-1(f(x)) = x for all x in the domain of f, and f(f-1(y)) = y for all y in the range of f.

What is the domain and range of a function?

The domain of a function is the set of all possible input values (or arguments) for the function. The range of a function is the set of all possible output values that the function can produce.

What is the difference between a one-to-one function and an onto function?

A one-to-one function is a function where each element in the domain maps to a unique element in the range. In other words, there are no two distinct elements in the domain that map to the same element in the range. An onto function (also called a surjective function) is a function where every element in the range is mapped to by at least one element in the domain.

What is the composition of functions?

The composition of functions is a new function that is created by applying one function to the output of another function. If f(x) and g(x) are two functions, then the composition of f and g, denoted by f o g, is defined as f o g(x) = f(g(x)). In other words, the output of g(x) is used as the input to f(x).

What is a polynomial function?

A polynomial function is a function of the form f(x) = an xn + an-1 xn-1 + … + a1 x + a0, where an, an-1, …, a0 are constants and n is a non-negative integer. The degree of a polynomial function is the highest power of x that appears in the function. For example, the function f(x) = 3x2 – 2x + 1 is a polynomial function of degree 2.

What is the domain of a function?

The domain of a function is the set of all possible inputs (or arguments) for the function. It is the set of all x-values for which the function is defined.

What is the range of a function?

The range of a function is the set of all possible outputs (or values) for the function. It is the set of all y-values that are produced by applying the function to the elements in the domain.