Math Homework: Functions, Sequences, Limits, Series, Geometry Problems

This document contains the solutions to a math homework assignment. The assignment covers several key concepts including determining the domain of functions, understanding odd functions, analyzing the graphical representation of functions, and evaluating the truth of mathematical statements. It also explores geometric proofs involving similar triangles and the lengths of line segments within circles. Furthermore, the assignment delves into the properties of functions, including one-to-one functions and their proofs. Lastly, the solutions provide analysis of the convergence and divergence of series using various tests. The assignment includes detailed explanations and justifications for each solution, providing a comprehensive understanding of the mathematical concepts involved.

MATHS
[DATE]

Question 1
(a) Domain
Domain of a given function is the argument value at which the function is well defined and
real.
Non-negative value for radicals
or
Merge overlapping intervals
Function is undefined points
Solution
Domain =
Interval Notation =
1

(b) If g(x) is an odd function which means it would be surely symmetric about the origin
Also,
Example:
(c) Graph of function
It can be seen that for every value of x in domain, there would be two values of y and hence,
it can be said that it is not a function of x.
2

(d) True
If
Let,
Now,
Which comes contradiction and hence, is TRUE.
(e) False
This sequence is bounded but has two limit and thus, it is divergent. Hence, it is FALSE
(f) False
Let
It is increasing sequence &
Therefore, it is convergent and hence, it is FALSE.
3

Question 2
(a) Need to prove are similar tringles
Radius of circle is a which means
In
Now,
Through, A-A-A similarity,
Here, are similar tringles (Proved).
4

(b) Need to prove that length of AE and BE are and
In
(Proved)
(Proved)
In
(Proved)
Question 3
(a) Graph
5

It is apparent from the graph that for different values of x, the function would have different
image and hence, function f(x) = one to one
(b)
Here,
Here, is increasing function so f(x) is termed as one to one function.
(c) Let
6

Here,
Hence, it can be concluded that function f(x) is one to one.
(d) Let
Now,
For different values of x, the function must possess different values and hence, f(x) would be
one to one.
(e) Graph
7

Domain: The function has no undefined points nor domain constraints. Hence, the domain
would be
Interval notation:
Range: The set of values of the dependent variable for which a function is defined.
Vertex of is (0,1)
Interval notation:
Question 4
(a) Converge or diverge
8

Converge
(b) Converge or diverge
Converge
(c) Converge or diverge
9

Diverge
10

(g)
(h)
11