Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#9
Question
a)Express in the form
, where
and
are constants.
The function f is defined by for
.
b)Find an expression for and state the domain of
.
The function is defined by
for
.
c)For the case where k = −1, solve the equation .
d)State the largest value of possible for the composition
to be defined.
Solution
a)
We are given that;
We use method of “completing square” to obtain the desired form. We complete the square for the terms which involve .
We have the algebraic formula;
For the given case we can compare the given terms with the formula as below;
Therefore, we can deduce that;
Hence, we can write;
To complete the square, we can add and subtract the deduced value of ;
b)
We are given that;
for
To find the inverse of a given function we need to write it in terms of
rather than in terms of
.
We have found in (a) that the given function can be written as;
Since given function is defined for , that means
is not possible.
Hence;
Interchanging ‘x’ with ‘y’;
Domain and range of a function become range and domain, respectively, of its inverse function
.
Domain of a function Range of
Range of a function Domain of
Therefore, if we find range of , then we can find domain of
.
Finding range of a function :
·Substitute various values of from given domain into the function to see what is happening to y.
·Make sure you look for minimum and maximum values of y by substituting extreme values of from given domain.
Therefore, domain of can be found from range of
;
We are given that function is defined for ; hence, domain of
is;
To find the range of , we substitute the least value of domain
in the function;
For ;
Hence range of is;
Range of a function Domain of
Hence, domain of ;
c)
The function f is for can be written as.
The function is defined by
for
when
.
We are required to solve the equation;
For a composite function , the domain D of
must be chosen so that the whole of the range of
is included in the domain of
. The function
, is then defined as
,
.
Therefore, range of will be domain of
.
Therefore, to find domain of we need to find range of
.
We are given that for
when
.
Finding range of a function :
·Substitute various values of from given domain into the function to see what is happening to y.
·Make sure you look for minimum and maximum values of y by substituting extreme values of from given domain.
Hence, range of ;
It is evident that domain of is;
Hence, we can conclude that only solution of the equation is;
d)
The function f is for can be written as.
The function is defined by
for
.
For a composite function , the domain D of
must be chosen so that the whole of the range of
is included in the domain of
. The function
, is then defined as
,
.
Since for
, for composite function
to be valid, we need to restrict range of
to domain of
.
We are given that domain of is;
Therefore, range of is;
Since for
;
For the range of
will remain within domain of
and hence, the composite function
will be valid.
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