**Problem Set based on previous three parts:**

**I) **Solve the inequalities in the following exercises expressing the solution sets as intervals or unions of intervals. Also, graph each solution set on the real line:

a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)

**II) **Quadratic Inequalities:

Solve the inequalities in the following exercises. Express the solution sets as intervals or unions of intervals and graph them. Use the result as appropriate.

(a) (b) (c) (d) (e) (f) (g) (h)

**III) **Theory and Examples:

i) Do not fall into the trap . For what real numbers a is the equation true? For what real numbers is it false?

ii) Solve the equation: .

iii) *A proof of the triangle inequality: *

Give the reason justifying each of the marked steps in the following proof of the triangle inequality:

…..why ?

….why ?

….why?

….why ?

iv) Prove that for any numbers a and b.

v) If and , what can you say about x?

vi) Graph the inequality:

**Questions related to functions:**

**I) **Find the domain and range of each function:

a) (b) (c)

**II) **Finding formulas for functions:

a) Express the area and perimeter of an equilateral triangle as a function of the triangle’ s side with length s.

b) Express the side length of a square as a function of the cube’s diagonal length d. Then, express the surface area and volume of the cube as a function of the diagonal length.

c) A point P in the first quadrant lies on the graph of the function . Express the coordinates of P as functions of the slope of the line joining P to the origin.

**III) **Functions and graphs:

Graph the functions in the questions below. What symmetries, if any, do the graphs have?

a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

**IV) **Graph the following equations ad explain why they are not graphs of functions of x. (a) (b)

**V) **Graph the following equations and explain why they are not graphs of functions of x: (a) (b)

**VI) **Even and odd functions:

In the following questions, say whether the function is even, odd or neither.

a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

**Sums, Differences, Products and Quotients:**

In the two questions below, find the domains and ranges of , , , and .

i) , (ii) ,

In the two questions below, find the domains and ranges of , , and

i) ,

ii) ,

**Composites of functions:**

- If , and , find the following: (a) (b) (c) (d) (e) (f) (g) (h)
- If and , find the following: (a) (b) (c) (d) (e) (f) (g) (h)
- If , , and , find formulas or formulae for the following: (a) (b) (c) (d) (e) (f)
- If , , and , find formulas or formulae for the following: (a) (b) (c) (d) (e) (f)

**Let **, , , and . Express each of the functions in the questions below as a composite involving one or more of f, g, h and j:

a) (b) (c) (d) (e) (f) (g) (h) (i) (k) (l) (m)

**Questions:**

a) , , find

b) , , find

c) , , find .

d) , , find

e) , , find .

f) , , find .

**Reference: Calculus and Analytic Geometry, G B Thomas. **

NB: I have used an old edition (printed version) to prepare the above. The latest edition may be found at Amazon India link:

Regards,

Nalin Pithwa