ordered pair

Examples of ordered pair in the following topics:

• Graphs of Linear Inequalities

  • If it is $>$ or $<$, then use a dotted or dashed line, since ordered pairs found on the line would result in a false statement.
  • Shading indicates all the ordered pairs in the region that satisfy the inequality.  
  • For example, if the ordered pair is in the shaded region, then that ordered pair makes the inequality a true statement.
  • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
  • All possible solutions are shaded, including the ordered pairs on the line, since the inequality is $\leq$ the line is solid.  

• Graphs of Equations as Graphs of Solutions

  • In an equation where $x$ is a real number, the graph is the collection of all ordered pairs with any value of $y$ paired with that real number for $x$.
  • For example, to graph the equation $x-1=0,$ a few of the ordered pairs would include:
  • After substituting the rest of the values, the following ordered pairs are found:
  • After graphing the ordered pairs and connecting the points, we see that the set of (infinite) points follows this pattern:
  • Construct the graph of an equation by finding and plotting ordered-pair solutions

• Piecewise Functions

  • Allowing $y=f(x)$, where $f(x)=|x|$, some ordered pair examples of $(x,|x|)$ are:
  • Some ordered pair examples are:
  • Those points satisfy the first part of the function and create the following ordered pairs:
  • For the middle part (piece), $f(x)=3$ (a constant function) for the domain $1ordered pairs are:
  • For the last part (piece), $f(x)=x$ for the domain $x>2$, a few ordered pairs are:

• Graphical Representations of Functions

  • The ordered pairs normally stated in linear equations as $(x,y)$, in function notation are now written as $(x,f(x))$.
  • Next, substitute these values into the function for $x$, and solve for $f(x)$ (which means the same as the dependent variable $y$):  we get the ordered pairs:
  • Next, plug these values into the function, $f(x)=x^{3}-9x$, to get a set of ordered pairs, in this case we get the set of ordered pairs:

• Introduction to Domain and Range

  • We can also tell by the set of ordered pairs given in this mapping that it is a function because none of the $x$-values repeat: $(-1,1),(1,1),(7,49),(0.5,0.25)$; since each input maps to exactly one output.  
  • We can also tell this mapping, and set of ordered pairs is a function based on the graph of the ordered pairs because the points do not make a vertical line.  
  • Let's look at this mapping and list of ordered pairs graphed on a Cartesian Plane.
  • In order to find the domain of a function, if it isn't stated to begin with, we need to look at the function definition to determine what values are not allowed.
  • This mapping or set of ordered pairs is a function because the points do not make a vertical line.  

• Introduction to Systems of Equations

  • In order for a linear system to have a unique solution, there must be at least as many equations as there are variables.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
  • We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.
  • An independent system has exactly one solution pair $(x, y)$.

• One-to-One Functions

  • A list of ordered pairs for the function are:
  • The ordered pairs $(-2,4)$ and $(2,4)$ do not pass the definition of one-to-one because the element $4$ of the range corresponds to to $-2$ and $2$.
  • Notice also, that these two ordered pairs form a horizontal line; which also means that the function is not one-to-one as stated earlier.

• Equations in Two Variables

  • Each solution is an ordered pair and can be written in the form $(x, y)$.
  • There are thus an infinite number of ordered pairs that satisfy the equation.$$
  • Note that the ordered pair $(3, 10)$ tells us that $x = 3$ and $y = 10$.

• Graphing Equations

  • After creating a few $x$ and $y$ ordered pairs, we will plot them on the Cartesian plane and connect the points.
  • Through the same arithmetic as above, we get the ordered pairs $(10,0)$ and $(-10,0)$.
  • The line continues on to infinity in each direction, since there is an infinite series of ordered pairs of solutions.

• Graphing Inequalities

  • Recall that for a linear equation in two variables, ordered pairs that produce true statements when substituted into the equation are called "solutions" to that equation.
  • We say that an inequality in two variables has a solution when a pair of values has been found such that substituting these values into the inequality results in a true statement.
  • Recall that, in order to graph an equation, we can substitute a value for one variable and solve for the other.
  • The resulting ordered pair will be one solution to the equation.