4 types of relations (1 Viewer)

Jigmey

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Was away for the lesson so am unsure on how to do these questions. Also don’t know what the | x-3| means. Any help is appreciated.
 

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fan96

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That's called the absolute value function, and it's defined by



Basically, you only want the magnitude of the number, not its sign.

For example,






If , then either or .

A useful identity to know is

 

Jigmey

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That's called the absolute value function, and it's defined by



Basically, you only want the magnitude of the number, not its sign.

For example,






If , then either or .

A useful identity to know is

so y = |x-3| would become y = x+3?
 

fan96

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so y = |x-3| would become y = x+3?
Unfortunately it's not that simple. You can't "break up" the absolute value function like that.

If , then .

You can try graphing these to visualise the effect of the absolute value function.
 

TheShy

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You could go onto desmos and mess around with the absolute function to graphically gain an understanding of it
 

CM_Tutor

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That's called the absolute value function, and it's defined by

So, expanding on the idea:



FYI, the absolute value function is a special case of what is called the modulus (only appears in Extension 2) but which is also the basis for the use of the same vertical line symbols in vectors. can be interpreted as the distance along the number line from the origin to x. Being a distance, it cannot be negative. Applying this idea, can be interpreted as the distance along the number line from 3 to x. This idea extends beyond one dimension, so that means the distance from P to Q and thus the length of the vector .
 

CM_Tutor

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Similar questions (so you can do the originals yourself):

1(a). Explain, with an example using a y-value, why each function is many-to-one.

(i)
(ii)
(iii)
(iv)

(b) Hence classify each relation below.

(i)
(ii)
(iii)
(iv)

1(a) If a graph passes the vertical line test - that is, that all values of x in the domain produce a single value of y, though the y-values can be different for each x-value - and thus the graph represents a function. That function is "many-to-one" if the graph fails the horizontal line test, and thus that there exists at least one value of y that leads to multiple (more than one) values of x. If the graph passes the horizontal line test then the function is "one-to-one." All four questions in 1(a) are functions as, in each case, choosing any value of x produces only one value of y.

(i) Consider :



For this function, the value leads to two possible values of , and thus means the function is many-to-one instead of one-to-one.

(ii) Consider :



For this function, the value leads to two possible values of , and thus means the function is many-to-one instead of one-to-one.

(iii) Consider :



For this function, the value leads to three possible values of , and thus means the function is many-to-one instead of one-to-one.

(iv) Consider :



For this function, the value leads to two possible values of , and thus means the function is many-to-one instead of one-to-one.

(b) Swapping x and y produces a relation that swaps the vertical and horizontal line results, so all of the relations in (b) will be one-to-many.


2(a). By solving for x, show that each function is one-to-one. (this method works only if x can be made the subject. Then, the relation is one-to-one if there is never more than one answer, and many-to-one otherwise.)

(i)
(ii)
(iii)

(b) Hence classify each relation below.

(i)
(ii)
(iii)

2(a)(i) and (b)(i):


Each value of x leads to one value of y, and each value of y leads to only one value of x. In other words, the graph passes both the horizontal and vertical line tests, and we have a one-to-one function - as must the relation in (b)(i).

2(a)(ii) and (b)(ii):


Each value of x leads to one value of y, and each value of y leads to only one value of x. In other words, the graph passes both the horizontal and vertical line tests, and we have a one-to-one function - as must the relation in (b)(ii).

2(a)(iii) and (b)(iii):


Each value of x leads to one value of y, and each value of y leads to only one value of x. In other words, the graph passes both the horizontal and vertical line tests, and we have a one-to-one function - as must the relation in (b)(iii).
 

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