Below is sample of QUANTITATIVE ANALYSIS STUDY NOTES as per July 2021 syllabus

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**TOPIC 1**

**FUNCTIONS**

Functions are often representatives of real phenomena or events. Functions therefore are models. Obtaining a function to act as a model is commonly the key to understanding business in many areas.

Functions may be represented formulae. There are a number of common ways in which functions are presented and used. We shall consider functions given formulae, since this provides a natural context for explaining how a function works.

**Functions of one variable**

If you get a job that pays Ksh 200 per hour, the amount of money M that you earn depends on the number of hours (h) that you work, and the relationship is given a simple function:

The formula *M=200h* shows that the money *M* that you earn depends on the number of hours worked. We say that *M* is a function of *h*.

In this context, *h* is a variable whose value we may not know until the end of the week. Once the value of h is known, the formula M=200h can be used to calculate the value of M. To emphasise that M is a function of h, it is common to write M=M(h),so that M(h)=700h.

**Example 1**

if you work 30 hours, then the function M(30) is the money you would earn. To calculate the amount earned, you need to replace the formula 30. Thus, M = M(30) = 700×30 = sh. 21,000.

**Note:**

- It is important to remember that
*h*is measured in hours and*M*is measured given is Kenya shillings. The function/formula is not useful unless you state in words the units you are using. - We would also use different letters or symbols for the variables. Whatever letter/symbol used, it is critical that you explain in word what they mean.

**Functional notation and substitution**

We normally write functions as f(x) and read as “function f of x”. We could use other letters such as g, p, H or h and write the function of x as g(x), p(x), H(x) or h(x).

Given f(x) = 3x +5, the value of this function f(x) when x=0 is written as f(0). The values are obtained substituting.

**Example 2**

**a) Given f(x) = 3x + 5,**find i) f(0), ii) f(2) and iii) f(-2).**b) Given that h(x) =2×2 +5x +3,**find i) h(0). ii) h(-2) and iii) h(3).**c) Given that G(t) = 2 +3t – 5t2 +t3,**find i) g(-1) and g(2).**d) If h(x) = 5 – 2x**, find the value of x for which the function is zero.

- a) i) f(0) =3(0) + 5 = 0 + 5 = 5
- ii) f(2) =3(2)+5 = 6 + 5=11

iii) f(-2) = 3(-2) + 5 = -6 +5 = -1

- b) i) h(0) = 2(0)2+ 5(0)+3 = 3
- ii) h(-2) = 2(-2)2 + 5(-2) +3 =1

iii) h(3) = 2(3)2+ 5(3)+3 = 36

- c) i) G(-1) = 2+3(-1)-5(-1)2-2(-1)3 = -4
- ii) G(2) = 2+3(2) – 5(2)2 – 2(2)3 = -28

- d) h(x) = 5-2x = 0

-2x = – 5 x = 2.5

**Example 2. 3**

- a) If f(x) = 5x -3. find f(x+2).
- b) If y = g(x) = 3x – 7. find g(y) in terms of x.

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