Graphing - Part I

@

Prepared for Gina. L

Domain and Range

@

Domain - set of "input (x)" value where function is defined.

Range  - set of "output (y)" value where function is defined.

@

Question 1.1

Sketch and find the domain / range for following functions.

a) y = x2b)  y = (x + 2)  ; c) y = |x|  ; d) y = 1/x  ;  e) y = sin(x) ;   f) y = log (x + 1)

@

@

Solution 1.1

a) y = x2

Domain (-, ) 

Range [0, )

@

@

b) y = (x + 2)

Domain [-2, )

Range [0, )

@

c) y = |x|

Domain (-, )

Range [0, )

@

d) y = 1/x

Domain (-, 0)   (0, ) 

Range (-, 0)   (0, )

@

e) y = sin (x)

Domain (-, )

Range [-1, 1]

@

f) y = log (x + 1)

Domain (-1, )

Range (-, )

@

@

Even / Odd function

@

A function can be even, odd or neither.  

If it's an even function, than it have a symmetry relation as...

                         f(x) = f(−x)

@

If it's an odd function, than it have a symmetry relation as...

                         f(x) = −f(−x)

@

Otherwise, its neither.

@

Question 2.1

Determine whether the following function is even, odd or neither.

a) f1(x) = xb) f2(x) = x3  ;  c) f3(x) = |x|  ;  d) f4(x) = x  ;  e) f5(x) = x4 + 2x3

@

Solution 2.1

a) f1(x) = x2

Step 1: Get f1(−x) 

f1(−x) = (−x)2

          = x2

@

Step 2: Compare f(−x) with f(x).

f1(x) = x2 = f1(−x) 

@

Step 3: Result

Answer: f1(x) is an even function.

@

b) f2(x) = x3

@

f2(−x) = (−x)3

          = −x3

@

f2(x) = x3 = −(−x3) = − f2(−x)

f2(x) = −f2(−x)

@

Answer: f2(x) is an odd function.

@

c) f3(x) = |x|

@

f3(−x) = |−x|

          = |x|

@

f3(x) = |x| = f3(−x)

f3(x) = f3(−x) 

@

Answer: f3(x) is an even function.

@

d) f4(x) = x

@

f4(−x) = (−x)

          

f4(x) = x  (−x) = f4(−x)

f4(x) = x  −(−x) = −f4(−x)

@

Answer: f4(x) is neither.

@

e) f5(x) = x4 + 2x3

@

f5(−x) = (−x)4 + 2(−x)3

          = x4 − 2x3

@

f5(x) = x4 + 2x3  x4 − 2x3 = f5(−x)

f5(x) = x4 + 2x3  −x4 + 2x3 = −f5(−x)

@

Answer: f5(x) is neither.

@

... to be continue

@

@

Written by Kevin Tang (Mar 9, 2010)

[back]

Copyright 2010 - KEVKEVWORLD.NET