function

Examples of function in the following topics:

• Inverse Trigonometric Functions

  • Each trigonometric function has an inverse function that can be graphed.
  • To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
  • However, the sine, cosine, and tangent functions are not one-to-one functions.
  • As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.
  • The arcsine function is a reflection of the sine function about the line $y = x$.

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

• Increasing, Decreasing, and Constant Functions

  • As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.
  • We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
  • In terms of a linear function $f(x)=mx+b$, if $m$ is positive, the function is increasing, if $m$ is negative, it is decreasing, and if $m$ is zero, the function is a constant function.
  • In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.  
  • A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.  

• Composition of Functions and Decomposing a Function

  • Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
  • The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
  • The resulting function is known as a composite function.
  • Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function.
  • Practice functional composition by applying the rules of one function to the results of another function

• One-to-One Functions

  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
  • One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.  
  • A list of ordered pairs for the function are:
  • The graph of the function $f(x)=x^2$ fails the horizontal line test and is therefore NOT a one-to-one function.  
  • If a horizontal line can go through two or more points on the function's graph then the function is NOT one-to-one.

• Sine and Cosine as Functions

  • As with the sine function, we can plots points to create a graph of the cosine function.
  • A periodic function is a function with a repeated set of values at regular intervals.
  • Specifically, it is a function for which a specific horizontal shift, $P$, results in a function equal to the original function:
  • For even functions, any two points with opposite $x$-values have the same function value.
  • The sine and cosine functions are periodic, meaning that a specific horizontal shift, $P$, results in a function equal to the original function:$f(x + P) = f(x)$.

• Hyperbolic Functions

• Graphical Representations of Functions

  • Function notation, $f(x)$ is read as "$f$ of $x$" which means "the value of the function at $x$."  
  • This function is that of a line, since the highest exponent in the function is a $1$, so simply connect the three points.
  • The graph for this function is below.
  • The function is linear, since the highest degree in the function is a $1$.  
  • The degree of the function is 3, therefore it is a cubic function.

• Inverses of Composite Functions

  • A composite function represents, in one function, the results of an entire chain of dependent functions.
  • In mathematics, function composition is the application of one function to the results of another.
  • In general, the composition of functions will not be commutative.
  • A composite function represents in one function the results of an entire chain of dependent functions.
  • If $f$ is an invertible function with domain $X$ and range $Y$, then

• The Existence of Inverse Functions and the Horizontal Line Test

  • Recognize whether a function has an inverse by using the horizontal line test