NDA 2010 Math Review

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Questions and Answers

What is the range of the function f(x) = cos(x/3)?

[-1, 1] (correct)

(-3, 3)

(-1/3, 1/3)

(1/3, -1/3)

What is the domain of the function f(x) = sin^{-1}[log2(x/2)]?

[-1, 4] (correct)

None of these

[1, 4]

[-4, 1]

Which of these describes the range of the function f(x) = |x + 2|?

{-1, 1}

{0, 1}

R - {-2}

R (correct)

If f(x) = x + 2, what is f(3)?

5 Signup and view all the answers

What is the range of the function f(x) = (x^2 + 1)/(x)?

(-2, ∞) Signup and view all the answers

For the function f(x) = x + 2, what will f(0) yield?

2 Signup and view all the answers

If f(x) = 3x^2, what is f(1)?

3 Signup and view all the answers

What is the property of the graph of an odd function?

It is symmetric about the origin. Signup and view all the answers

What can be concluded if the composition of two functions g and f, written as gof, is one-one?

f must be one-one. Signup and view all the answers

What is a necessary condition for the LCM of a rational number and an irrational number to exist?

There is no such condition; the LCM does not exist. Signup and view all the answers

Which statement about the derivative of an odd function is correct?

The derivative is also an odd function. Signup and view all the answers

If f(x) is defined as both even and odd, what can be stated about f(x)?

f(x) must equal zero for all x. Signup and view all the answers

Given two finite sets A and B with m and n elements respectively, how many one-one functions can be formed from A to B if n < m?

0 Signup and view all the answers

What does the function f(x) + f(−x) represent if f(x) is any function?

An even function. Signup and view all the answers

If A and B are two different sets with disjoint elements m and n respectively, how many total mappings from A to B can be established?

mn Signup and view all the answers

What is the domain of the function f described?

The set of all real numbers, R Signup and view all the answers

What defines the range of the function f?

The set of all possible outputs of f Signup and view all the answers

Which of the following statements is true regarding the outputs of the function f?

Each input must be linked to exactly one output Signup and view all the answers

If y = f(x) = 2 - cos(3x), under what condition does y not represent a function of x?

If there is an input x for which more than one output y exists Signup and view all the answers

What can be concluded about the relationship between the codomain and the range of function f?

The range is always a subset of the codomain Signup and view all the answers

Which of these statements is NOT a requirement for a relation to be considered a function?

Each input can correspond to multiple outputs Signup and view all the answers

For the function defined as f(x) = 2 - cos(3x), what is a possible range of f?

[1, 3] Signup and view all the answers

In the function f(x), what does the absence of an image for any input x signify?

The input does not belong to the domain of f Signup and view all the answers

For the function defined by $f(x) = ext{cos}^2(x) + ext{sin}^4(x)$, what is the range of values for $f(R)$?

[0, 1] Signup and view all the answers

If $f(x) = 2x + 7$ and $g(x) = x^2 + 7$, what values of $x$ satisfy $g(f(x)) = 8$?

0 Signup and view all the answers

What transformation does the equation $f(x) = kf(\frac{200x}{100 + x^2})$ imply about $f(x)$?

It scales $f$ by a factor of $k$. Signup and view all the answers

If $f(x) = x|x|$, which statement about the function $f$ is accurate?

It is neither one-to-one nor onto. Signup and view all the answers

Which value of $k$ satisfies the equation $e^{f(x)} = \frac{10 + x}{10 - x}$ for $x \in (-10, 10)$?

0.8 Signup and view all the answers

When examining the functions $f(x) = 2x + 3$ and $g(x) = x^2 + 7$, which is correct about $g(f(x)) = 8$?

It leads to a quadratic equation. Signup and view all the answers

What type of function is $f(x)$ if $f'(x)$ is even?

f is an even function. Signup and view all the answers

Which range corresponds with the conditions $0 < x < rac{3}{2}$ and $x = y = R$?

R+ Signup and view all the answers

What kind of function is f defined by the set f = {(1, 1), (2, 1), (3, 0)}?

One-to-one but not onto Signup and view all the answers

For which values of x does the equation f(g(x)) = 25 hold, given f(x) = 2x + 3 and g(x) = x^2 + 7?

±2 Signup and view all the answers

What is the output of f(2002) if f(x) is defined as f(x) = cos^2(x) + sin^4(x)?

1 Signup and view all the answers

What is the domain of the function defined as f(x) = sin^(-1)(log2(x))?

{ x : 1 ≤ x ≤ 2 } Signup and view all the answers

If f(x) = 1/(x + 2) + 1/(2x - 4) for x > 2, what is f(11)?

1 Signup and view all the answers

What is the value of f(x) if f(x) is defined as f(x) = cos(log(x))?

Dependent on the value of x Signup and view all the answers

In the equation f(g(x)) = 25, which mathematical operation does g(x) = x^2 + 7 represent?

Polynomial transformation Signup and view all the answers

How would you classify f(x) = 2|x| in terms of its behavior?

Increasing for x > 0 and decreasing for x < 0 Signup and view all the answers

Study Notes

Function Ranges and Definitions

Range of ( f(x) = \cos(x/3) ) is ([-1, 1]).
The composition of functions: for ( f: R \to R ) and ( g(x) = x + 3 ), if ( (f \circ g)(x) = (x + 3)^2 ), then ( f(-3) ) can be determined.

Domain of Functions

Domain of ( f(x) = \sin^{-1}[\log_2(x/2)] ) is defined; among choices: ([1, 4]) is one valid selection.
Domain of ( f(x) = \sin^{-1}(\log_2 x) ) in real numbers is a set constrained by logarithmic properties.

Composition and Inverses

If ( f(x) = 2x + 3 ) and ( g(x) = x^2 + 7 ), solving ( g(f(x)) = 8 ) leads to specific values of ( x ).
Let ( f(x) = x + 3; f(-3) ) seeks output using inverse properties.

Range of Functions

The range of ( f(x) = \frac{|x + 2|}{x^2 + 1} ) is ( \mathbb{R} ) excluding (-2).
For ( f(x) = \cos(\log_e x) ), the combined output satisfies specific conditions.

Properties of Functions

A function is defined as a mapping from set X to Y, connecting each element in X to one element in Y.
(-1 \leq \cos(3x) \leq 1) ensures the output for ( f(x) ) remains bounded.

Functional Relationships

If ( f(x) ) is injective (one-to-one), then ( g(f(x)) ) retains that property.
Functions are defined as even if ( f(x) ) symmetry about the y-axis holds.

Function Transformations

For transformations, if ( f ) maps real numbers avoiding certain outputs, it affects the domain significantly, constraining areas like ( (−10, 10) ).

Relationships in Sets

In finite sets ( A ) and ( B ) of different sizes, counting functions can determine how many one-to-one or onto mappings exist, based on element count.
Special cases exist for functions returning zero, being both odd and even.

Miscellaneous Properties

Calculations for compositions, such as ( (f \circ f)(x) = x ), require careful definition and function identity.
Symmetry in graphs aids in determining function properties effectively; odd functions reflect about the origin while even ones reflect about axes.

General Definitions

( f(x) + f(-x) ) is even; ( f(x) - f(-x) ) is odd.
Conditions for ( g \circ f ) to inherit properties depend on the constituent functions being one-to-one or onto.

Practical Application

Examples illustrate how to derive function outputs through definitions, compositions, consistency in output behavior, and constraints based on real-number properties.

Studying That Suits You

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Description

Test your understanding of trigonometric functions and composite functions with this quiz based on the NDA 2010 exam. Key questions include the range of the cosine function and the domain of inverse sine functions. Perfect for math enthusiasts and NDA exam aspirants!