Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Functions - characteristics - even and odd functionss.
Test Yourself 1 - Solutions.


 

Even functions 1.Show that f(x) = -3x2 -1 is an even function. 2. Show that f(x) = 2x6 + 3x4 + 14 is an even function.
  3. Show that 4.
  5. What type of value for n would be needed if the function f(x) = x4 + xn is an even function? 6. Why can we conclude by inspection that
f(x) = x2 + x + 1 cannot be an even function?
Odd functions 7. Show that f(x) = x3 - 3x is an odd function 7. Show that f(x) = 3x7 + x5 - 11x is an odd function.
  9. Why can we conclude by inspection that
f(x) = x9+ x5 + 1 cannot be an even function?		 10. A function f(x) = xn + x3n is an odd function.

Suggest a possible value for n.

Answer.n = +1 or -1 or any odd number.
Graphs - given. 11. 12.
  13. 14.
Graphs. 15. A function is partly defined as f(x) = 2x - 1 for x > 0.
It is also known that the function f(x) is an odd function.

(i) Complete the definition for f(x) for x ≤ 0.

(ii) Draw a neat sketch of y = f(x).

Answer.f(x) = 3x - 1 for x > 0.
f(x) = 0 for x = 0.
f(x) = 3x + 1 for x < 0.

16. (i) Sketch the function y = x2 for 0 ≤ x ≤ 2 and write down its range.

(ii) If we know that the function in (i) is part of a larger function f(x) which is odd and whse domain is
-2 ≤ x ≤ 2, sketch the function f(x) on a second diagram.
 

17. (i) Draw a graph showing f(x) = 2 (x ≠ 0).

(ii) On the same axes, draw the continuation of the function given that it is odd.

  
  19. The original function:

 

17 (cont'd) The continuation of the function if it is odd.

Notice the pattern on the left as the red pattern on the right is rotated 180o about the origin.
     
     
Even and/or odd functions.

23. Is the function f(x) = x2 + 2x - 8

odd or even?

 Find a function g(x) if f(x) is an odd function and

f(x) = x2 for x > 0

f(x) = g(x) for x < 0.

Answer.g(x) = -x2.
 

25.

26. Prove is an even function.
  27. Determine if the function

is an even, odd or neither function.

 28.
Manipulation

29. If and g(x) = x2, find the composite function f(g(x)).

State the domain and range of this composite function.