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Math Worksheet 1– FUNCTION versus RELATION
Relations
A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as:
Mapping Diagram of Relation Graph of Relation y is not a function of x (x = 0 has multiple outputs)
Functions
A function is a relation in which each input x ( domain ) has only one output y ( range ).
To check if a relation is a function, given a mapping diagram of the relation, use the following criterion:
- If each input has only one line connected to it, then the outputs are a function of the inputs.
- The Vertical Line Tests for Graphs To determine whether y is a function of x, given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x. If you can draw a vertical line that goes through 2 points, y is not a function of x. This is called the vertical line test.
In the following graphs:
y is a function of x (passes vertical line trest) y is not a function of x (fails vertical line test)
Function notation
There is a special notation, that is used to represent this situation: if the function name is f , and the input name is x ,then the unique corresponding output is
called f ( x ) (which is read as " f of x ".)
We can also use letters: g (x), h (x) or simply y
Question: What does the function notation g (7) represent? Answer: the output from the function g when the input is 7
Question: Suppose f ( x ) = x + 2. What is f (3)? Answer: f (3) = 3 + 2 = 5 (simply substitute number 3 for the variable x)
Question: Suppose f ( x ) = x + 2. What is f ( x +5)? Answer: f ( x +5) = ( x + 5) + 2 = x + 7
Operations with functions
Given f ( x ) = 3 x + 2 and g ( x ) = 4 – 5 x , find ( f + g )( x ), ( f – g )( x ), ( f × g )( x ), and ( f / g )( x ).
( f + g )( x ) = f ( x ) + g ( x ) = [3 x + 2] + [4 – 5 x ] = 3 x – 5 x + 2 + 4 = – 2 x + 6
( f – g )( x ) = f ( x ) – g ( x ) = [3 x + 2] – [4 – 5 x ] = 3 x + 5 x + 2 – 4 = 8 x – 2
( f × g )( x ) = [ f ( x )][ g ( x )] = (3 x + 2)(4 – 5 x ) = 12 x + 8 – 15 x^2 – 10 x = – 15 x^2 + 2 x + 8
Homework
State the domain and range of each relation. Then determine whether each relation is a function
Graph each relation or equation and determine the domain and range.
Find each value if f ( x ) = − 5 x + 2 and g ( x ) = - 2 x + 3.
7. f (3) 8. f (-4) 9. g ( −1 2)
10. f (-2) 11. g (-6) 12. f ( m - 2) 13. Use the functions below to perform the following operations:
f(x) = 2x g(x) = x – 2 h(x) = x^2 k(x) = x/
k(x) x f(x)
g(x) - h(x)
f(x) - k(x) h(x) + k(x) f(x) ÷ k(x) g(x) x h(x)
