Math 1314
2.2 More on Functions and Their Graphs
Notation of functions.
f(x) or y =f(x) is a notation of a function f.
x is called independent variable.
y is called dependent variable.
Examples:
1. Let f(x) = 2x – 7. Find:
a/ f(3) =
b/ f(- 2)
c/ f(3/2)
d/ f(a)
e/ f(a + h)
f/
f ( x  h)  f ( x)
h
Section 2.2 Notes
Math 1314
2. Let g(x) = x2 – 2x – 3. Find:
a/ g(1) =
b/ g(- 3)
c/ g(0)
d/ g(a)
e/ g(a + h)
f/
g ( x  h)  g ( x)
h
Section 2.2 Notes
Math 1314
Section 2.2 Notes
3. Let f ( x) 
1
. Find:
2x
a/ f(5) =
b/ f(- 2)
c/ f(1/2)
d/ f(a)
e/ f(a + h)
f/
f ( x  h)  f ( x)
h
Math 1314
Section 2.2 Notes
Even and Odd Functions
Even Functions
Let f(x) be a function on the domain D. Then f is even if f(x) = f(-x) for all x in the domain of f.
-
Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis,
meaning that its graph remains unchanged after a reflection about the y-axis.
-
Examples of even functions are
f(x) = x2, f(x) = x4 + 1
Below is a graph of even function y = f(x) = x4 + 1, it is symmetric over the y-axis.
Odd Functions
Let f(x) be a function on the domain D. Then f is odd if f(x) = -f(-x) for all x in the domain of f.
-
Geometrically speaking, the graph face of an odd function is symmetric with respect to the origin,
meaning that its graph remains unchanged after a rotation of 180 degrees about the origin (or after a
reflection about the y-axis then the x-axis)
-
Examples of even functions are f(x) = x3, f(x) = x5 + 5x.
Below is a graph of the odd function y = f(x) = x3, it is symmetric over the origin.
Math 1314
Section 2.2 Notes
Note: To verify a function is neither even nor odd, we just need to give a counter-example i.e. f(x) ≠ -f(-x) for
some value of x.
Example: Use the definition to verify whether each given function is even or odd, or neither.
1. f(x) = x3 – 6x
2. g(x) = x4 – 2x2
3. h(x) = x2 + 2x + 1
Math 1314
4.
f ( x)  x 1  x 2
Section 2.2 Notes
Math 1314
Piecewise Functions
Section 2.2 Notes
A piecewise function is a function defined by more than one equation.
 2 x  4 if x  3

Example: f ( x)   2 if  3  x  2
 x  2 if x  2

is a piecewise function.
Evaluating and Graphing Piecewise Functions.
Example: Given the piecewise functions. For each function, evaluate the function at the given values of x. Then
graph the function.
1.
f(-2) =
f(0) =
f(3) =
 2 x  1 if
f ( x)  
5  x if
x0
x0
Math 1314
2.
f(5) =
f(0) =
Section 2.2 Notes
 x 1
f ( x)  
 3x  9
if
x2
if
x2
Math 1314
3.
f(-2) =
f(0) =
f(3) =
 3x
f ( x)   2
x
if
if
x 1
x 1
Section 2.2 Notes
Math 1314
4.
f(-2) =
f(0) =
f(3) =
Section 2.2 Notes
2x  4

f ( x )  2
 -x  1

if
x  -1
if - 1  x  2
if
x2