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).