Graphs and Functions in Mathematics

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Questions and Answers

What is the value of f(x) = [x] when x = 2.5?

2.5

2 (correct)

Which property of the greatest integer function indicates that [x + I] equals [x] + I for integer I?

Property (iii)

Property (iv)

Property (ii) (correct)

Property (i)

If φ(x) = 3.5, according to the properties of the greatest integer function, what can be inferred about [φ(x)]?

[φ(x)] = 3 (correct)

[φ(x)] = 2

[φ(x)] = 5

[φ(x)] = 4

What is the fractional part of x when x = 4.7?

0.7 Signup and view all the answers

What does [−x] equal if x is an integer?

[−x] = −[x] Signup and view all the answers

What is the nature of the function y = tan x as x approaches its asymptotes?

It tends to infinity Signup and view all the answers

At which values of x does the function y = tan x have its vertical asymptotes?

$rac{3 heta}{2}$ for all $ heta$ Signup and view all the answers

What is the range of the function y = tan x?

-∞ to +∞ Signup and view all the answers

How does the function y = tan x behave as x increases through its defined intervals?

It increases strictly Signup and view all the answers

Which of the following describes the asymptotic behavior of y = tan x near the points x = ±$rac{ heta}{2}$?

The function approaches +/- infinity Signup and view all the answers

What is the primary focus of the book 'Play with Graphs'?

To teach students graph drawing techniques for solving problems Signup and view all the answers

Who is the author of 'Play with Graphs'?

Amit M. Agarwal Signup and view all the answers

Which of the following is NOT mentioned as a goal of the book?

To provide complex mathematical theories Signup and view all the answers

What is indicated about the ease of understanding graphs in the book?

Drawing different types of graphs can be understood by average students Signup and view all the answers

What type of audience is 'Play with Graphs' primarily aimed at?

Engineering aspirants Signup and view all the answers

How long has the author been guiding students?

For over a decade Signup and view all the answers

In what format is the publisher's address presented in the book?

A detailed physical address is provided Signup and view all the answers

What issue does the author recognize about existing resources on mathematical problems?

There is a need for a more effective approach to understanding them Signup and view all the answers

What is the relationship between the functions $y=x$ and $y=-x$ as indicated in the figures?

They intersect at the origin. Signup and view all the answers

Why can't trigonometric functions be directly inverted?

They are many-one functions across their domain. Signup and view all the answers

What is a necessary step to make the inverse of a trigonometric function a valid function?

Restrict the domain to a valid interval. Signup and view all the answers

What does the term 'asymptote' refer to in the context of the provided graphs?

A line that the graph approaches but never touches. Signup and view all the answers

What is the behavior of the function $y = a^x$ when $a > 1$ and $x > 1$?

The function is strictly increasing. Signup and view all the answers

Which property of the sine function is highlighted as a reason for its inversion?

It spans all values between -1 and 1. Signup and view all the answers

What occurs at the points (c, c) and (–c, –c) in the context of the asymptotes?

They correspond to the intersections of the asymptotes and the function. Signup and view all the answers

Which statement is true for the function $y = a^x$ when $0 < a < 1$?

The function decreases with the increase in $x$. Signup and view all the answers

What can be concluded about the function $y = 2^x$, $y = 3^x$, and $y = 4^x$ for $x > 1$?

They are strictly increasing, with $2^x < 3^x < 4^x$. Signup and view all the answers

Which of the following statements is true regarding inverse trigonometric functions?

They are defined by limiting the domain of original functions. Signup and view all the answers

Which function is specifically mentioned as being used in creating an inverse trigonometric function?

Sine function Signup and view all the answers

How does the function $y = a^x$ behave when $0 < x < 1$ and $a > 1$?

The function decreases, with $4^x < 3^x < 2^x$. Signup and view all the answers

What defines the range of a logarithmic function?

All real numbers. Signup and view all the answers

What indicates that a function is invertible in the context of $y = a^x$?

It is strictly increasing or strictly decreasing. Signup and view all the answers

For which value of $a$ is the function $y = a^x$ characterized as decreasing?

0 < a < 1 Signup and view all the answers

What happens to the values of $y = a^x$ as $x$ approaches infinity when $a > 1$?

The values approach positive infinity. Signup and view all the answers

Study Notes

Introduction of Graphs

Graphs are a visual representation of data, used to understand relationships between different variables.
Graphs are useful for analyzing trends, patterns, and relationships in data, making complex information easier to comprehend.

Greatest Integer Function

The greatest integer function, denoted by [x], gives the greatest integer less than or equal to x.
For example, [3.2] = 3, [-2.7] = -3, and [5] = 5.

Fractional Part of Function

The fractional part of x, denoted by {x}, is the difference between x and its greatest integer value.
For example, {3.2} = 0.2, {-2.7} = 0.3, and {5} = 0.
Fractional part functions are expressed as y = {x}.

### Tangent Function

The tangent function, denoted by y = tan(x), is a periodic function with a period of π.
Its graph has vertical asymptotes at x = (2n + 1)π/2, where n is an integer.
Tangent functions increase strictly from -∞ to +∞ as x increases.

Exponential Functions

Exponential functions are of the form y = a^x, where a is a positive constant and a ≠ 1.
The function is increasing when a > 1 and decreasing when 0 < a < 1.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions.
The general form is y = log_a(x) where a is a positive constant and a ≠ 1.
Logarithmic functions are defined for positive real numbers, with a domain of all real positive numbers and a range of all real numbers.

Inverse Functions

A function is invertible if and only if it is one-to-one, meaning each input value corresponds to a unique output value.
Inverse functions reverse the mapping of the original function.

Inverse Trigonometric Functions

Trigonometric functions are not invertible, meaning each output value can correspond to multiple input values.
To make trigonometric functions invertible, their domains are restricted, creating inverse trigonometric functions: sin⁻¹(x), cos⁻¹(x), tan⁻¹(x), etc.

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Description

This quiz explores key concepts in mathematics, focusing on different types of graphs and functions, including the greatest integer function, fractional part function, tangent function, and exponential functions. Understanding these concepts is essential for analyzing data and interpreting mathematical relationships effectively.