function

Examples of function in the following topics:

• Expressing Functions as Power Functions

  • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
  • A power function is a function of the form $f(x) = cx^r$ where $c$ and $r$ are constant real numbers.
  • Polynomials are made of power functions.
  • Functions of the form $f(x) = x^3$, $f(x) = x^{1.2}$, $f(x) = x^{-4}$ are all power functions.
  • Describe the relationship between the power functions and infinitely differentiable functions

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
  • They are one of the simplest functional forms.
  • Linear functions may be confused with affine functions.
  • However, the term "linear function" is quite often loosely used to include affine functions of the form $f(x)=mx+b$.
  • A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$.

• Inverse Functions

  • An inverse function is a function that undoes another function: For a function $f(x)=y$ the inverse function, if it exists, is given as $g(y)= x$.
  • Inverse function is a function that undoes another function: If an input $x$ into the function $f$ produces an output $y$, then putting $y$ into the inverse function $g$ produces the output $x$, and vice versa. i.e., $f(x)=y$, and $g(y)=x$.
  • If $f$ is invertible, the function $g$ is unique; in other words, there is exactly one function $g$ satisfying this property (no more, no less).
  • Not all functions have an inverse.
  • A function $f$ and its inverse $f^{-1}$.

• Continuity

  • A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
  • A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.
  • Otherwise, a function is said to be a "discontinuous function."
  • A continuous function with a continuous inverse function is called "bicontinuous."
  • This function is continuous.

• Further Transcendental Functions

  • A transcendental function is a function that is not algebraic.
  • Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
  • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
  • Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
  • Bottom panel: Graph of sine function versus angle.

• Inverse Functions

  • An inverse function is a function that undoes another function.
  • Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.
  • Not all functions have an inverse.
  • Let's take the function $y=x^2+2$.
  • We can check to see if this inverse "undoes" the original function by plugging that function in for $x$:

• The Derivative as a Function

  • If every point of a function has a derivative, there is a derivative function sending the point $a$ to the derivative of $f$ at $x = a$: $f'(a)$.
  • This function is written $f'(x)$ and is called the derivative function or the derivative of $f$.
  • Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions.
  • It is only defined on functions:
  • At the point where the function makes a jump, the derivative of the function does not exist.

• Derivatives of Exponential Functions

  • The derivative of the exponential function is equal to the value of the function.
  • Functions of the form $ce^x$ for constant $c$ are the only functions with this property.
  • The slope of the graph at any point is the height of the function at that point.
  • The rate of increase of the function at $x$ is equal to the value of the function at $x$.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

• Inverse Trigonometric Functions: Differentiation and Integration

  • The inverse trigonometric functions are also known as the "arc functions".
  • There are three common notations for inverse trigonometric functions.
  • The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable.
  • Thus each function has an infinite number of antiderivatives.
  • Note that some of these functions are not valid for a range of $x$ which would end up making the function undefined.

• Vector-Valued Functions

  • Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
  • A three-dimensional vector valued function requires three functions, one for each dimension.
  • This is a three dimensional vector valued function.
  • This can be broken down into three separate functions called component functions:
  • This function is representing a position.