# 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

The function

**c)**For the case where k = −1, solve the equation

**d)**State the largest value of

**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;

To find the inverse of a given function

We have found in (a) that the given function can be written as;

Since given function is defined for

Hence;

Interchanging ‘x’ with ‘y’;

Domain and range of a function

Domain of a function

Range of a function

Therefore, if we find range of

Finding range of a function

·Substitute various values of

·Make sure you look for minimum and maximum values of y by substituting extreme values of

Therefore, domain of

We are given that function is defined for

To find the range of

For

Hence range of

Range of a function

Hence, domain of

**c)
**

The function f is for

The function

We are required to solve the equation;

For a composite function

Therefore, range of

Therefore, to find domain of

We are given that

Finding range of a function

·Substitute various values of

·Make sure you look for minimum and maximum values of y by substituting extreme values of

Hence, range of

It is evident that domain of

Hence, we can conclude that only solution of the equation

**d)
**

The function f is for

The function

For a composite function

Since

We are given that domain of

Therefore, range of

Since

For

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