functions and relations
Functions and Relations A very important concept with which we need to spend some time to define and understand, is the Domain and Range. The Domain is the set of numbers that are fed into the formula ( x ) and the Range is the set of outputs ( y ). Look at the following example y = - 4x + 1: If we choose the following x values: x {2; 5; 7;} then the corresponding y values will be: For x = 2: y = -4(2) + 1 = - 7 For x = 5: y = -4(5) + 1 = - 19 For x = 7: y = -4(7) + 1 = - 27 The formula is applied to the x values, and the Domain is therefore x {2; 5; 7} and the y values that we calculate, will be the Range: y { -7; -19; -27} We have encountered the case of two inputs yielding the same 2 output y x , but we could also run into a formula that will produce more than one output for a single specific input. Observe the formula y² = x If this is solved for y, we will get y x , which means, that, if you take x = 4, the y-values will be 2 AND – 2!
Functions
We will then have to distinguish between these three types of portrayals i.e. Where each element of the Domain has got only one portrayal in the Range
Where more than one element of the Domain portray the same element in the Range
Where a single element from the Domain portrays more than one element in the Range We can then stipulate the following definition: All portrayals are RELATIONS Only single Range relations are Functions In normal language this means that the first two types will be functions, but the third type will not be Functions
Example : Determine whether y = x² + 1, x {1; 2; 3} and y R, is a function or a relation Also provide the Domain and Range. For x = 1 For x = 2 For x = 3
=> => =>
y = (1)² + 1 = 2 y = (2)² + 1 = 5 y = (3)² + 1 = 10
Every x portrays only one y, and therefore it is a function. Domain: x {1; 2; 3} Range : y {2; 5; 10} If you are given a graph, with the instructions to determine whether it is a function or a relation, you could make use of the vertical line test. This means, simply, that you draw a vertical line through the graph. If this line intersects the graph more than once, at any point, it will not represent a function.
Relation
Function Functions
This technique of drawing lines, whether it is vertical or horizontal, could also assist you in determining the Domain and Range: Wherever these lines do not intersect the graph anymore, will be values outside of the Domain or Range, as in the following example:
Domain: x R; x ≠ 0
Range: y R; y ≠ 0
Functions
Functions
We will then have to distinguish between these three types of portrayals i.e. Where each element of the Domain has got only one portrayal in the Range
Where more than one element of the Domain portray the same element in the Range
Where a single element from the Domain portrays more than one element in the Range We can then stipulate the following definition: All portrayals are RELATIONS Only single Range relations are Functions In normal language this means that the first two types will be functions, but the third type will not be Functions
Example : Determine whether y = x² + 1, x {1; 2; 3} and y R, is a function or a relation Also provide the Domain and Range. For x = 1 For x = 2 For x = 3
=> => =>
y = (1)² + 1 = 2 y = (2)² + 1 = 5 y = (3)² + 1 = 10
Every x portrays only one y, and therefore it is a function. Domain: x {1; 2; 3} Range : y {2; 5; 10} If you are given a graph, with the instructions to determine whether it is a function or a relation, you could make use of the vertical line test. This means, simply, that you draw a vertical line through the graph. If this line intersects the graph more than once, at any point, it will not represent a function.
Relation
Function Functions
This technique of drawing lines, whether it is vertical or horizontal, could also assist you in determining the Domain and Range: Wherever these lines do not intersect the graph anymore, will be values outside of the Domain or Range, as in the following example:
Domain: x R; x ≠ 0
Range: y R; y ≠ 0
Functions
