FUNCTIONS
The Domain of a Function

Example:

The domain is the set of all real numbers
for which the expression is defined as a
rea...
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 funct...
Vertical Line Test: A graph is a function if every
vertical line intersects the graph at most one
point.

Example:

Find f...
Composition of Functions
Now let’s consider a very important way of combining two functions to get a new
function.
Given t...
Example:
Inverse Functions

Horizontal Line Test

One-to-One Function

A function f is one-to-one if and only if
A function f with ...
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 i...
Functions by mstfdemirdag
Functions by mstfdemirdag
Functions by mstfdemirdag
Functions by mstfdemirdag
Functions by mstfdemirdag
Functions by mstfdemirdag
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Functions by mstfdemirdag

  1. 1. FUNCTIONS
  2. 2. The Domain of a Function Example: The domain is the set of all real numbers for which the expression is defined as a real number. f(x) = 2x + 3 D=R g(x) = 2x + 3 D=R Example: f(x) = g(x) Example: 1 f (x ) = x −4 D = R – {4} f (x ) = x D = R+ g(x) = x f (x ) = 5 f (x ) = 2 − x 2x − 6 x f (x ) = 2 f (x ) = x 2 − 4 x −9 f (x ) = Equal Functions Two functions are equal if and only if their expressions and domains are equal. 2 x x D=R f(x) ≠ g(x) D = R – {0} Even and Odd Function A function is called even if f(-x) = f(x) A function is called odd if f(-x) = -f(x) Example: State whether each of the following functions are even or odd function. f(x) = 3x2 + 4 h(x) = 2x3 g(x) = x m(x) = x3 – 1
  3. 3. 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.
  4. 4. Vertical Line Test: A graph is a function if every vertical line intersects the graph at most one point. Example: Find f + g, f - g, f·g, and f/g Operations on Functions: Homework: Page 53 check yourself 13 Homework: Page 53 check yourself 13
  5. 5. Composition of Functions Now let’s consider a very important way of combining two functions to get a new function. Given two functions f and g, the composite function f o g (also called the composition of f and g) is defined by (f o g)(x) = f( g(x) )
  6. 6. Example:
  7. 7. Inverse Functions Horizontal Line Test One-to-One Function A function f is one-to-one if and only if A function f with domain D and range R is a every horizontal line intersects the graph of f in at most one point. 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: Example: Check whether the following functions are one-to-one. f(x) = 3x + 1 g(x) = x2 - 3 h(x) = 1 - x
  8. 8. 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) The two graphs are reflections of each other through the line y = x , or are symmetric with respect to this line. if and only if x = g(y) How to find inverse of a function Solve the equation x = f(y) for y. f(x) = 3x + 7

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