A relation as a function provided there is exactly one output for each input. It is NOT a function if at least one input has more than one output A function is to have a limit if it has a two-sided limit. A graph provides a visual method of determining the limit of a function. If the function has a limit as x approaches a, the branches of the graph will approach the same y-coordinate near x=a from the left and the right. Consider functions f :A (⊂ X) → Y where X and Y are metric spaces. Important cases are 1. f :A (⊂ R) → R, 2. f :A (⊂ Rn) → R, 3. f :A (⊂ Rn) → Rm Sometimes we can sketch the domain and the range of the function, perhaps also with a coordinate grid and its image.Sometimes we can represent a function by drawing a vector at various points in its domain to represent f(x). We can represent a real-valued function by drawing the level sets (contours) of the function. In other cases the best we can do is to represent the graph, or the domain and range of the function, in a highly idealized manner..Sometimes we can represent a function by drawing a vector at various points in its domain to represent f(x).

1. Diagrammatic
Representation of
Functions
MA. SHARON M. ARTILLO
MILE Math

2. Functions
Page 116
•A relation as a function provided there is
exactly one output for each input.
•It is NOT a function if at least one input has
more than one output

3. INPUT
(DOMAIN)
OUTPUT (RANGE)
FUNCTION
MACHINE
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
Functions
ONE INPUT CAN HAVE ONLY ONE
OUTPUT

4. A function is to have a limit if
it has a two-sided limit. A
graph provides a visual
method of determining the
limit of a function. If the
function has a limit as x
approaches a, the branches of
the graph will approach the
same y-coordinate near x=a
from the left and the right.

7. Important cases are
 1. f :A (⊂ R) → R,
 2. f :A (⊂ Rn) → R,
 3. f :A (⊂ Rn) → Rm
Consider functions f :A (⊂ X) → Y where X and Y are
metric spaces.

8. Functions can be quite complicated, and we
should be careful not to be misled by the
simple features of the particular graphs which
we are able to sketch.
We can represent a function by its graph

9. 1. f :A (⊂ R) → R,

12. Sometimes we can sketch the domain and the
range of the function, perhaps also with a
coordinate grid and its image.

13. Sometimes we can represent a
function by drawing a vector at
various points in its domain to
represent f(x).

14. We can represent a real-valued function by drawing
the level sets (contours) of the function. In other
cases the best we can do is to represent the graph,
or the domain and range of the function, in a
highly idealised manner..

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