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
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