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Relations and Functions

Relations

Functions and relations
Relations and functions map a domain (x values) to a range (y values). A function is a special type of relation.

Relations describe the manner in which an independent variable maps to a dependent variable.

A line is a relation between any x value and a single y value in the Cartesian coordinate system. The data could also be represented by a table with two columns. The pairs of values are called 'Ordered Pairs': the Domain (input) is the largest set of values which map to the Range (output or image) set of values.

A relation is a function $f$ if:

- $f$ acts on all elements of the domain

- $f$ pairs each element of the domain with one and only one element of the range

In Cartesian graphs of relations and functions, traditionally $y = f(x)$. The relation can also be expressed as $f: x ↦ f(x)$, where $f$ is a function that maps $x$ into $f(x)$.

The independent variable is $x$ (also known as the argument of the function), and the dependent variable is y.

Functions

A function is a correspondence (mapping) between two sets X and Y in which each element of set X maps to (corresponds with) exactly one element of set Y.

Vertical line test

A relation is a function if a vertical line drawn anywhere on a graph of the relation does not cross the curve more than once.

e.g. a vertical line drawn through $f(x) = x^2$ crosses no more than once (function), but $f(x) = ±√x$ has two intersections (not a function).

A vertical line parallel to the y-axis has more than one (in fact infinite!) y-values for one x-value. A vertical line is a relation but not a function.

A horizontal line parallel to the x-axis has the same y-value for any value of x. It is a relation AND a function.

Graph
Graph of the quadratic $y = x^2$

The relation $y = x^2$ has one y-value for two x-values, except at the minimum or maximum, where there is one for one. This is a function, so may be expressed as $f(x) = x^2$.

Graph
Graph of the relation $y = ±√x$

The relation $y^2 = x$, however, has two y-values for each x, so is not a function.

Asymptotes and Discontinuities

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Now consider the equation: $g(x) = 1/{(x^2-1)}$



What are its domain and range?

If the denominator of a fraction is zero, there is no solution. As the denominator approaches zero, the value of the function gets very large (tends to infinity).

This function therefore has three asymptotes: x = -1, x = +1, y = 0.

Domain: ]-∞, -1[ ∪ ]-1, +1[ ∪ ]+1, +∞[

The graph is discontinuous for values of g(x) between -1 and 0.

Range: ]-∞, -1] ∪ ]0, +∞[

Even and odd functions

A function is even, if for all $x$ in the domain, $-x$ is in the domain, and $f(x) = f(-x)$ for all values of $x$.

Even functions are symmetrical about the y-axis. cos$(x)$ is an example of an even function.

A function is odd, if for all $x$ in the domain, $-x$ is in the domain, and $f(-x) = -f(x)$ for all values of $x$.

Odd functions are NOT symmetrical about the y-axis. sin$(x)$ is an example of an odd function.

one-to-one and one-to-many

Remember that a relation which has more than one value of y for each value of x is not a function.

However, a y value may have more than one value of x. This is a one-to-many relationship. An example is $y = x^2$.

If there is only one value of y for each value of x, such as in $y = x$, then the function is one-to-one.

A horizontal line test will reveal whether there is more than one x value for one y value.

Example: even and odd

Mathematics question
$h(x) = x - 3x^3 + 1/2x^5$
Is $h(x)$ even or odd? Solve algebraically.
Is $h(x)$ one-to-one or many-to-one?

If $h(x)$ is even, then $h(x) = h(-x)$ for all values of $x$.

$h(-x) = -x + 3x^3 - 1/2x^5 = -(x - 3x^3 + 1/2x^5) ≠ h(x)$
∴ $h(x)$ is not even.

If $h(x)$ is odd, then $h(-x) = -h(x)$ for all values of $x$.

$h(-x) = -x + 3x^3 - 1/2x^5 = -(x - 3x^3 + 1/2x^5) = -h(x)$

∴ $h(x)$ is odd and many-to-one.

Rational functions

A rational function is a function of the form $f(x)={g(x)}/{h(x)}$, where $g$ and $h$ are polynomials.

A rational function of the form $f(x) = k/{x-b}$ has a vertical asymptote at $x=b$, since this takes the denominator to zero.

A rational function of the form $f(x) = {ax+b}/{cx+d}$ forms a hyperbola. A hyperbola has a horizontal asymptote at $y=a/c$, and a vertical asymptote at $x=-d/c$.

Hyperbola
Hyperbola: a rational function which is the quotient of two linear equations

To find the horizontal asymptote of a hyperbola, we rearrange the equation to find when $x$ has no solution:

$y= {ax+b}/{cx+d}$

$y(cx+d)= {ax+b}$

$x(yc -a) = b - dy$

$x= {b-dy}/{cy-a}$

When $y=a/c$ there is a horizontal asymptote ($x$ approaches infinity as $y$ approaches $a/c$).

Conic Sections
Conic Sections: slicing a cone forms the shapes of four special functions

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