 # Relations and Functions

## 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 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 \$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: slicing a cone forms the shapes of four special functions

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