Functions by mstfdemirdag

FUNCTIONS

The Domain of a Function

The domain is the set of all real numbers for which the expression is defined as a real number.

Example:

D = R – {4}

D = R+

Equal Functions

Two functions are equal if and only if their expressions and domains are equal.

Example:

f(x) = 2x + 3 D = R

g(x) = 2x + 3 D = Rf(x) = g(x)

Example:

g(x) = x D = R

D = R – {0}f(x) ≠ g(x)

Even and Odd Function

A function is called even if f(-x) = f(x)

A function is called odd if f(-x) = -f(x)

State whether each of the following functions are even or odd function.

Example:

f(x) = 3x2 + 4

g(x) = x

h(x) = 2x3

m(x) = x3 – 1

What is use of even and odd functions?

Graph of a function is symmetric respect to y-axis if it is even.

Graph of a function is symmetric respect to origin if it is odd.

Example: Classify whether the following functions are even or odd.

Vertical Line Test: A graph is a function if every vertical line intersects the graph at most one point.

Operations on Functions:

Find f + g, f - g, f·g, and f/gExample:

Homework: Page 53 check yourself 13 Homework: Page 53 check yourself 13

Composition of Functions

Now let’s consider a very important way of combining two functions to get a newfunction.

Given two functions f and g, the composite function f o g (also called thecomposition of f and g) is defined by (f o g)(x) = f( g(x) )

Example:

Inverse FunctionsOne-to-One Function

A function f with domain D and range R is a one-to-one function if either of the following equivalent conditions is satisfied:(1) Whenever a≠b in D, then f(a) ≠ f(b) in R.(2) Whenever f(a) = f(b) in R, then a=b in D.

Example:

Check whether the following functions are one-to-one.

f(x) = 3x + 1

g(x) = x2 - 3

h(x) = 1 - x

Horizontal Line Test

A function f is one-to-one if and only if every horizontal line intersects the graph of f in at most one point.

Example:

Inverse Function

Let f be a one-to-one function with domain D and range R. A function g with domain R and range D is the inverse function of f, provided the following condition is true for every x in D and every y in R:

y = f(x) if and only if x = g(y)

The two graphs are reflections of each other through the line y = x , or are symmetric with respect to this line.

How to find inverse of a function

Solve the equation x = f(y) for y.

f(x) = 3x + 7