Algebra/Chapter 5/Functions

Algebra

See also: Calculus/Functions, Discrete Mathematics/Functions and relations

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Even and odd functions

Even functions

An even function is defined as a function $f$ such that $f(-x)=f(x)$.
Geometrically an even function can be defined as a function that exhibits a mirror image symmetry across the y-axis (the vertical line that passes through the origin).

An example of an even function is $h(x)=x^{2}$ because $f(5)=25=f(-5)$ and because $f(x)=x^{2}=f(-x)$ for all real numbers x.

Odd functions

An odd function is defined as a function $f$ such that $f(-x)=-f(x)$.
Geometrically an odd function can be defined as a function that exhibits a 180 degree rotational symmetry about the origin.

An example of an odd function is $f(x)=x^{3}$ because for all real numbers x, $f(x)=x^{3}=-((-x)^{3})=-f(-x)$ for example $f(2)=2^{3}=8=-((-2)^{3})=-(-8)=-((-2)^{3})=-f(-2)$

Composite function

A composite function $h$ can be defined as the composite of the two functions $f$ and $g$ and denoted as $h(x)=f(g(x))$ (read h of x is equal to f of g of x) or $h(x)=(f\circ g)(x)$.

Example:

Let $f(x)=2x+1$     $g(x)=5x-3$
 $h(x)=f(g(x))$
 $h(x)=f(5x-3)$
 $h(x)=2(5x-3)+1$
 $h(x)=10x-6+1$
∴$h(x)=10x-5$

Example:

Let $f(x)=-{\sqrt {16-x}}\,$     $g(x)=4x^{2}\,$
$(f\circ g)(x)=f(4x^{2})\,$
$(f\circ g)(x)=-{\sqrt {16-(4x^{2})}}\,$
$(f\circ g)(x)=-{\sqrt {4(4-x^{2})}}\,$
$(f\circ g)(x)=-{\sqrt {4}}{\sqrt {(4-x^{2})}}\,$
$(f\circ g)(x)=-2{\sqrt {4-x^{2}}}\,$
Domain: $-2\leq x\leq 2$
Range: $-4\leq y\leq 0$

Inverse function

The function $g$ is the inverse of the one-to-one function $f$ if and only if the following are true:

$g(f(x))=x\,$

$f(g(x))=x\,$

The inverse of function $f$ is denoted as $f^{-1}$ .

Geometrically $f^{-1}$ is the reflection of $f$ across the line $y=x$. Conceptually, using the box analogy, a function's inverse box undoes what the function's regular box does.

Example:

$f(x)=2x\,$
$f^{-1}(x)={\frac {1}{2}}x\,$

$f(f^{-1}(x))=f({\frac {1}{2}}x)\,$
$f(f^{-1}(x))=2({\frac {1}{2}}x)\,$
$f(f^{-1}(x))=x\,$


$f^{-1}(f(x))=f^{-1}(2x)\,$
$f^{-1}(f(x))={\frac {1}{2}}(2x)\,$
$f^{-1}(f(x))=x\,$

To find the inverse of a function, remember that when we use $f^{-1}(x)$ as an input to $f$ the result is $x$. So start by writing $x=f\left(f^{-1}(x)\right)$ and solve for $f^{-1}$

Example:

Suppose:$f(x)=2x+1\,$
Then $x=f\left(f^{-1}(x)\right)$
$x=2f^{-1}(x)+1\,$
$x-1=2f^{-1}(x)\,$
$f^{-1}(x)={\frac {x-1}{2}}\,$

The Domain of an inverse function is exactly the same as the Range of the original function. If the Range of the original function is limited in some way, the inverse of a function will require a restricted domain.

Example:

$f(x)={\sqrt {x-1}}\,$         
$x=f\left(f^{-1}(x)\right)$
$x={\sqrt {f^{-1}(x)-1}}\,$         
$x^{2}={\sqrt {f^{-1}(x)-1}}^{2}\,$         
$x^{2}=f^{-1}(x)-1$            
$f^{-1}(x)=x^{2}+1\,$            
The Range of $f(x)$ is $f(x)\geq 0$. So the Domain of $f^{-1}(x)$ is $x\geq 0$.

One-to-one function

A function that for every input there exists an output unique to that input.

Equivalently, we may say that a function $f$ is called one-to-one if for all $x,x'\in A,f(x)=f(x')$ implies that $x=x'$ where A is the domain set of f and both x and x' are members of that set.

Horizontal Line Test
If no horizontal line intersects the graph of a function in more than one place then the function is a one-to-one function.

Creating Functions

In the previous chapter we reviewed what you've already learned about mathematics: Numbers, Variables, and Relationships. We reviewed the types of numbers, the operations you can perform on numbers, the properties of these operations, and how these properties can allow you to write expressions, or if we know about enough the constraints on the expressions you can write equations and inequalities that define things that are true.

In the section above we've looked at the concept of a function. First we showed how to created equations with a function on one side of the equals operator and an expression on the other. Then we looked at more complicated ways to use function notation.

Once you get used to them functions give you a different way to look at math. When you think about math with numbers you are thinking about just one answer. When you think about math with functions you are looking for relationships and you are building mathematical models.

Give an example of a 3 X 3 square with a diagonal. What is the area of one of the triangles from the diagonal. Apply the Area function: l X w, and then the half function.