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Relations and Functions


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.


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 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 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

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: 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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John Herschel

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John Herschel is the son of William Herschel, and the nephew of Caroline Herschel, two famous astronomers. He continued his father's work, publishing enhanced catalogues of astronomical objects, but was also prolific in many other fields of science and technology, notably as a pioneer of photography.

John Herschel
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