one-to-one function

Examples of one-to-one function in the following topics:

• 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.
  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain.
  • One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.  
  • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
  • If a horizontal line can go through two or more points on the function's graph then the function is NOT one-to-one.

• Inverse Trigonometric Functions

  • For a one-to-one function, if $f(a) = b$, then an inverse function would satisfy $f^{-1}(b) = a$.
  • However, the sine, cosine, and tangent functions are not one-to-one functions.
  • In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods.
  • 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.
  • Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible.

• Sine and Cosine as Functions

  • Recall that the sine and cosine functions relate real number values to the $x$- and $y$-coordinates of a point on the unit circle.
  • Notice how the sine values are positive between $0$ and $\pi$, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between $\pi$ and $2\pi$, which correspond to the values of the sine function in quadrants III and IV on the unit circle.
  • As with the sine function, we can plots points to create a graph of the cosine function.
  • The points on the curve $y = \sin x$ correspond to the values of the sine function on the unit circle.
  • The points on the curve $y = \cos x$ correspond to the values of the cosine function on the unit circle.

• Increasing, Decreasing, and Constant Functions

  • The figure below shows examples of increasing and decreasing intervals on a function.
  • Look at the graph from left to right on the $x$-axis; the first part of the curve is decreasing from infinity to the $x$-value of $-1$ and then the curve increases.  
  • The curve increases on the interval from $-1$ to $1$ and then it decreases again to infinity.
  • The function $f(x)=x^3−12x$ is increasing on the $x$-axis from negative infinity to $-2$ and also from $2$ to positive infinity.  
  • The function is decreasing on on the interval: $ (−2, 2)$.  

• 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.
  • Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function.
  • In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure; i.e. when constituent functions are found to depend on approximately disjointed sets of variables.
  • Practice functional composition by applying the rules of one function to the results of another 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.
  • The functions $g$ and $f$ are said to commute with each other if $g ∘ f = f ∘ g$.
  • A composite function represents in one function the results of an entire chain of dependent functions.
  • The entire chain of dependent functions are the ingredients, drinks, plates, etc., and the one composite function would be putting the entire chain together in order to calculate a larger population at the school.

• 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.
  • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.
  • We can factor the denominator to find the singularities of the function:
  • However, for $x^2 + 2=0$ , $x^2$ would need to equal $-2$.

• Translations

  • A translation of a function is a shift in one or more directions.
  • To translate a function vertically is to shift the function up or down.
  • To translate a function horizontally is the shift the function left or right.
  • When $a$ is positive, the function is shifted to the right.  
  • When $a$ is negative, the function is shifted to the left.  

• Introduction to Exponential and Logarithmic Functions

  • Logarithmic functions and exponential functions are inverses of each other.
  • Lastly, as with all inverse functions, if we graph $f(x)=log_{b}x$  and $f^{-1}(x)=b^{x}$ on the same plane, the graphs will be symmetric across the line $y=x$.
  • That is, if we fold the plane over the line $y=x$, the two curves will lie on each other.
  • Another way of thinking about this is that if we generate points on the curve of $f(x)=log_{b}x$ we can find the points on the curve of  $f^{-1}(x)=b^{x}$ by interchanging the $x$ and $y$ coordinates of the points.
  • Further, a point $(t,u=b^t)$ on the graph of $f(x)$ yields a point $(u,t=log{_b}u)$ on the graph of the logarithm and vice versa.

• Stretching and Shrinking

  • Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase.
  • If the function $f(x)$ is multiplied by a value less than one, all the $y$ values of the equation will decrease, leading to a "shrunken" appearance in the vertical direction.  
  • If $b$ is greater than one the function will undergo vertical stretching, and if $b$ is less than one the function will undergo vertical shrinking.
  • If we want to vertically stretch the function by a factor of three, then the new function becomes:
  • If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.