# 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

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**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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