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High School Mathematics
8.2 Relations and Functions

A, B are two sets. f is a relation from A into B. If f is such that for every a A and there is a unique b B such that (a,b) f, then f is said to be a function from A into B.

If f is a function from A into B then we write it f:A B and we read this as f maps A into B.

A function is also called a mapping.
If f: A B is a mapping and (a,b) f, then we write it f(a) = b, f(a) is called the image of a.

A is called the domain of f and B is called the co-domain of f. The set of f(A) of all images of elements of A under the mapping f is called the range of f.

Example:

  • Continuity: A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.
  • Increasing functions: y = x
  • Decreasing functions: y = -x
  • Constant functions: x = 3 or y = 4
  • Even and Odd functions:
    • A function f is called even function if f(-x) = f(x) for all the x in the domain of f. The graph is symmetric with respect to y-axis.
    • A function f is called odd function if f(-x) = - f(x) for all the x in the domain of f. The graph is symmetric with respect to origin.
  • Identity function is defined by f(x) =x is increasing and continuous on its entire domain from - infinity to + infinity.
  • The square root function is defined by f(x) = sqrt(x) increases and is continuous on its entire domain from 0 to positive infinity.
  • The cube root function is defined by f(x) = cube-root(x) increases and is continuous on its entire domain from negative infinity to positive infinity.
  • The absolute value function is defined by f(x) = |x| decreases on the interval (-infinity, 0] increases on the interval [0, infinity) and continuous on its entire domain (- infinity, positive infinity)

Directions: Choose the correct answer. Write at least ten examples of relations and if they are functions or not functions. Also draw graph of increasing, decreasing, constant functions, even and odd functions, identity function, square root function, cube root function and absolute value function.
Q 1: R = {(1,a),(2,b),(3,b),(4,c)}, is a function.
Yes
No

Q 2: R = {(a,x),(a,y),(b,z),(c,z)}, is a function.
True
False

Q 3: The figure given below is an example of a function.

No
Yes

Q 4: The figure given below is an example of a function.

No
Yes

Q 5: f: A B is a mapping, in this B = _____.
Co-domain
Domain

Q 6: R = {(1,a),(1,b),(2,c),(3,c)}, is a function.
No
Yes

Question 7: This question is available to subscribers only!

Question 8: This question is available to subscribers only!


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