LimSome topics from maths 1its

Question 1: Identifying a Function from a Graph (from files image_98b860.png, image_98b592.png)

The Question: Which of the following may represent a function $f: \mathbb{R} \to \mathbb{R}$?

Core Concept: The Vertical Line Test

A graph represents a function if and only if every possible vertical line you can draw intersects the graph at at most one point. If any vertical line touches the graph in two or more places, it means one input ($x$-value) corresponds to multiple outputs ($y$-values), which violates the definition of a function.

Detailed Solution:

• Figure (a): A vertical line can be drawn (e.g., at $x=1$) that intersects this graph in two places (one with a positive y-value and one with a negative y-value). It fails the Vertical Line Test and is not a function.
• Figure (b): This is a parabola. Any vertical line drawn will intersect the graph exactly once. It passes the Vertical Line Test and is a function.
• Figure (c): This is a straight line. Any vertical line drawn will intersect the graph exactly once. It passes the Vertical Line Test and is a function.
• Figure (d): This is a cubic-like curve. Any vertical line drawn will intersect the graph exactly once. It passes the Vertical Line Test and is a function.

Final Answer: Figure (b), Figure (c), and Figure (d) may represent functions.

Question 2: Identifying a Linear Function (from file image_98b518.png)

The Question: Which of the following represents a linear function?

Core Concept: Linear Function A linear function is a type of polynomial where the highest power of the variable $x$ is 1. Its general form is $f(x) = mx + c$.

Detailed Solution:

• $f(x) = x+5$: Highest power of $x$ is 1. This is a linear function.
• $f(x) = 2x^2 + 5x + 10$: Highest power of $x$ is 2. This is a quadratic function.
• $f(x) = \ln(x^2+5)$: This is a logarithmic function, not a polynomial.
• $f(x) = x$: Highest power of $x$ is 1. This is a linear function.
• $f(x) = \frac{x}{3} + 3$: This can be rewritten as $f(x) = \frac{1}{3}x + 3$. The highest power of $x$ is 1. This is a linear function.

Final Answer: The linear functions are $f(x) = x+5$, $f(x) = x$, and $f(x) = \frac{x}{3} + 3$.

Question 3: Identifying a Quadratic Function (from file image_98b518.png)

The Question: Which of the following represents a quadratic function?

Core Concept: Quadratic Function A quadratic function is a type of polynomial where the highest power of the variable $x$ is 2. Its general form is $f(x) = ax^2 + bx + c$, where $a \neq 0$.

Detailed Solution:

• $f(x) = 7x+10$: Highest power is 1 (linear).
• $f(x) = 3x^2 + 9x + 199$: Highest power is 2. This is a quadratic function.
• $f(x) = \ln(2x^2+10)$: This is a logarithmic function.
• $f(x) = e^{2x}$: This is an exponential function.
• $f(x) = -x^2+1$: Highest power is 2. This is a quadratic function.

Final Answer: The quadratic functions are $f(x) = 3x^2 + 9x + 199$ and $f(x) = -x^2+1$.

Question 4: Domain of a Rational Function (from file image_98b518.png)

The Question: Consider the function $f(x) = \frac{x-1}{x^2 - 5x + 6}$. Then the number of points at which the function is not defined is __________.

Core Concept: Domain of a Rational Function A rational function (a fraction of polynomials) is not defined at any x-value that makes its denominator equal to zero, as division by zero is undefined.

Detailed Solution:

1. Identify the denominator: The denominator is $x^2 - 5x + 6$.
2. Set the denominator to zero: We need to find the roots of the equation $x^2 - 5x + 6 = 0$.
3. Solve the quadratic equation: We can solve this by factoring. We need two numbers that multiply to +6 and add to -5. The numbers are -2 and -3.
   • $(x-2)(x-3) = 0$
4. Find the roots:
   • $x-2 = 0 \implies x=2$
   • $x-3 = 0 \implies x=3$
5. Conclusion: The function is not defined at $x=2$ and $x=3$. There are two such points.

Final Answer: The number of points at which the function is not defined is 2.

Question 5: Identifying Sets (from file image_98b4ba.png)

The Question: Which of the following collections are sets?

Core Concept: Definition of a Set A set is a well-defined collection of distinct objects. “Well-defined” means that there is no ambiguity or matter of opinion in deciding whether an object belongs to the collection.

Detailed Solution:

• “Collection of irrational numbers.”: This is well-defined. Given any number, we can mathematically determine if it is irrational. This is a set.
• “Collection of rich people in India.”: This is not well-defined. The term “rich” is subjective and depends on individual opinion. There is no universal criterion. This is not a set.
• “Collection of movies directed by Rituparno Ghosh.”: This is well-defined. We can definitively check a film’s credits to see if he was the director. This is a set.
• “Collection of novels written by George Orwell.”: This is well-defined. An author’s bibliography is a matter of fact. This is a set.
• “Collection of good actors and actresses in India.”: This is not well-defined. “Good” is subjective and based on opinion.
• “Collection of good badminton players in India.”: This is not well-defined. “Good” is subjective.

Final Answer: The collections that are sets are:

• Collection of irrational numbers.
• Collection of movies directed by Rituparno Ghosh.
• Collection of novels written by George Orwell.

Question 6: Function Classification (from file image_98b4ba.png)

The Question: Which of the following option(s) is(are) true?

Core Concept: Hierarchy of Polynomial Functions Polynomials are a broad category of functions. Linear and quadratic functions are specific, simpler types of polynomials.

• Linear Function: A polynomial of degree 1.
• Quadratic Function: A polynomial of degree 2.

Detailed Solution:

• “All quadratic functions are polynomial functions.”: TRUE. By definition, a quadratic function is a polynomial of degree 2.
• “All polynomial functions are quadratic functions.”: FALSE. Polynomials can have any non-negative integer degree. For example, $f(x) = x^3$ is a polynomial but not quadratic.
• “All linear functions are polynomial functions.”: TRUE. By definition, a linear function is a polynomial of degree 1.
• “All quadratic functions are linear functions.”: FALSE. A quadratic function has degree 2, while a linear function has degree 1.

Final Answer: The true statements are:

• All quadratic functions are polynomial functions.
• All linear functions are polynomial functions.

Question 7: Increasing/Decreasing Intervals (from file image_98b1d0.png)

The Question: Consider the following statements about the above graph. Find the number of correct statements.

Core Concept: Reading a Graph When determining where a function is increasing or decreasing, we always read the graph from left to right.

• Increasing: The graph is going uphill (y-values are getting larger).
• Decreasing: The graph is going downhill (y-values are getting smaller). The points where the graph changes direction are called turning points.

Detailed Solution: From the graph, we can identify the turning points at approximately $x=-2$, $x=0$, and $x=2$.

• Interval $(-\infty, -2]$: The graph is going downhill. The function is decreasing.
• Interval $[-2, 0]$: The graph is going uphill. The function is increasing.
• Interval $[0, 2]$: The graph is going downhill. The function is decreasing.
• Interval $[2, \infty)$: The graph is going uphill. The function is increasing.

Now let’s evaluate the given statements:

1. “In the interval $(-\infty, 2]$, the function is increasing.” -> FALSE.
2. “In the interval $[-2, 2]$, the function is decreasing.” -> FALSE (it increases from -2 to 0).
3. “In the interval $[-2, 0]$, the function is increasing.” -> TRUE.
4. “In the interval $[0, 2]$, the function is decreasing.” -> TRUE.
5. “In the interval $[0, \infty)$, the function is increasing.” -> FALSE (it decreases from 0 to 2).
6. “In the interval $[2, 3]$, the function is increasing.” -> TRUE (since it’s increasing on $[2, \infty)$).

Final Answer: There are 3 correct statements.

Question 8: Comparing Growth Rates (from file image_98b158.png)

The Question: In the following Figure M2W1AQ 2, $C_1, C_2,$ and $C_3$ represent curves. Which of the following option(s) is(are) true?

Core Concept: Growth Rate The “fastest growth curve” is the one that gets steeper and reaches higher y-values more quickly as $x$ increases (moves to the right).

Detailed Solution:

Let’s observe the curves for positive x-values, especially for $x > 1$.

• Curve $C_1$ is the steepest. For any given $x > 1$, its y-value is the highest.
• Curve $C_2$ is in the middle.
• Curve $C_3$ is the least steep and has the lowest y-values for $x > 1$.

This means $C_1$ is growing fastest, followed by $C_2$, and then $C_3$. Now let’s evaluate the options:

• “Among these three curves, $C_1$ is the fastest growth curve.” -> TRUE.
• “Among these three curves, $C_2$ is the fastest growth curve.” -> FALSE.
• “Among these three curves, $C_3$ is the fastest growth curve.” -> FALSE.
• “$C_1$ is a faster growing curve than $C_2$.” -> TRUE.
• “$C_3$ is a faster growing curve than $C_2$.” -> FALSE.
• “$C_2$ is a faster growing curve than $C_3$.” -> TRUE.

Final Answer: The true statements are:

• Among these three curves, $C_1$ is the fastest growth curve.
• $C_1$ is a faster growing curve than $C_2$.
• $C_2$ is a faster growing curve than $C_3$.

Question 9: Always Increasing Functions (from file image_98b0d5.png)

The Question: Which of the following option(s) is(are) true?

• Any linear function is always an increasing function.
• Any quadratic function is always an increasing function.
• $e^x$ is always an increasing function.
• $\ln x$ is always an increasing function.

Detailed Solution:

• “Any linear function is always an increasing function.”: FALSE. A linear function $f(x)=mx+c$ is increasing only if its slope $m$ is positive. If $m$ is negative (e.g., $y=-2x$), it’s a decreasing function.
• “Any quadratic function is always an increasing function.”: FALSE. A quadratic function is a parabola ($\cup$ or $\cap$). It increases on one side of its vertex and decreases on the other. It is never always increasing.
• "$e^x$ is always an increasing function.": TRUE. The natural exponential function has a base $e \approx 2.718$, which is greater than 1. All exponential functions with a base greater than 1 are strictly increasing over their entire domain.
• "$\ln x$ is always an increasing function.": TRUE. The natural logarithm function is strictly increasing over its entire domain of $(0, \infty)$. As x gets larger, $\ln x$ gets larger.

Final Answer: The true statements are:

• $e^x$ is always an increasing function.
• $\ln x$ is always an increasing function.

Question 10: Analyzing Function Statements (from file image_98b0d5.png)

The Question: Consider the following statements. Find the number of correct statements.

Detailed Solution:

Let’s analyze each of the four statements.

1. "$f(x) = mx+c$ is an one-one function from $\mathbb{R}$ to $\mathbb{R}$, where $m$ is an arbitrary non-zero integer.": A non-zero slope means the line is not horizontal. Any non-horizontal line passes the Horizontal Line Test. Therefore, it is one-to-one. This statement is TRUE.
2. "$f(x) = x^2+bx+c$ is an onto function from $\mathbb{R}$ to $\mathbb{R}$, where $b$ and $c$ are arbitrary real numbers.": An onto (surjective) function from $\mathbb{R}$ to $\mathbb{R}$ must have a range of all real numbers. A quadratic function is a parabola with a minimum or maximum value. Its range is restricted (e.g., $[y_{vertex}, \infty)$). It is never all of $\mathbb{R}$. This statement is FALSE.
3. "$\frac{1}{e^x}$ is an unbounded function on $\mathbb{R}$.": The function is $f(x) = e^{-x} = (\frac{1}{e})^x$. This is an exponential decay function. As $x \to \infty$, $f(x) \to 0$. As $x \to -\infty$, $f(x) \to \infty$. A function is “unbounded” if its range goes to $\infty$ or $-\infty$. Since this function’s range is $(0, \infty)$, it is unbounded above but bounded below by 0. The statement is technically TRUE. However, if “unbounded” implies unbounded in both directions, it could be false. Let’s assume the standard definition.
4. “The function $g(x) = \frac{1}{1+e^x}$ gives a bijection from $\mathbb{R}$ to $(0, 1)$.”: A bijection means the function is both one-to-one (injective) and onto (surjective).
   • Domain: The denominator $1+e^x$ is never zero, so the domain is $\mathbb{R}$.
   • Range (to check if onto): As $x \to \infty$, $e^x \to \infty$, so $g(x) \to \frac{1}{\infty} \to 0$. As $x \to -\infty$, $e^x \to 0$, so $g(x) \to \frac{1}{1+0} \to 1$. The range is indeed $(0, 1)$. Since the range equals the specified codomain, the function is onto.
   • One-to-one: The derivative is $g’(x) = \frac{-e^x}{(1+e^x)^2}$, which is always negative. Since the function is strictly decreasing, it is one-to-one.
   • Since it is both one-to-one and onto for the given domain and codomain, it is a bijection. This statement is TRUE.

Final Answer: There are 3 correct statements.

Question 11: Injectivity and Surjectivity (from file image_98b0d5.png)

The Question: Consider the function $f(x) = x + \frac{1}{x}$. Which of the following options are correct?

Core Concept:

• Injective (One-to-one): No two different inputs give the same output.
• Surjective (Onto): The range of the function is equal to the specified codomain.

Detailed Solution:

Let’s analyze the function $f(x) = x + \frac{1}{x}$.

• “f is an injective function if the domain of f is $\mathbb{R} \setminus {0}$.”:

  • FALSE. Let’s find a counterexample. Let $x_1 = 2$. Then $f(2) = 2 + \frac{1}{2} = 2.5$.
  • Let $x_2 = \frac{1}{2}$. Then $f(\frac{1}{2}) = \frac{1}{2} + \frac{1}{1/2} = 0.5 + 2 = 2.5$.
  • Since two different inputs (2 and 1/2) give the same output, the function is not injective on this domain.

• “f is an injective function if the domain of f is $\mathbb{N}$.”:

  • TRUE. For the natural numbers ${1, 2, 3, …}$, let’s check if $x_1 + 1/x_1 = x_2 + 1/x_2$ implies $x_1=x_2$. On the interval $[1, \infty)$, the function is strictly increasing. Since $\mathbb{N}$ is a subset of this interval, different natural number inputs will give different outputs.

• “f is an injective function if the domain of f is $\mathbb{Q} \setminus {0}$.”:

  • FALSE. The same counterexample from the first point works, since 2 and 1/2 are both rational numbers.

• “f is an injective function if the domain of f is $\mathbb{Z} \setminus {0}$.”:

  • FALSE. Let $x_1 = 2$ and $x_2 = -2$. $f(2)=2.5$, $f(-2)=-2.5$. What about the original counterexample? 2 and 1/2 are not both in Z. Let’s check for counterexamples in Z. The function is increasing for $|x| \ge 1$. So for positive integers, it’s injective, and for negative integers, it’s injective. But $f(x)$ is never an integer for $|x| > 1$. Let’s test if $f(x_1)=f(x_2)$ for integers $x_1 \neq x_2$. The function is injective on this domain. Let’s re-verify. x+1/x. For positive integers 1, 2, 3... the values are 2, 2.5, 3.33.... For negative integers -1, -2, -3... the values are -2, -2.5, -3.33.... No values repeat. This statement is TRUE.
  • Correction: Let’s re-evaluate the previous answers based on this. The first option is definitely false. The second option on N is true. The third on Q is false. The fourth on Z{0} is true. This is an MSQ.

• “f is a surjective function if the domain of f is $\mathbb{N}$ and codomain of f is $\mathbb{Q}$.”:

  • FALSE. The range of f on $\mathbb{N}$ is ${2, 2.5, 3.33…, 4.25, …}$. Can we produce every rational number? No. For example, we can never produce an output of 1. To get 1, we would need $x+1/x=1 \implies x^2-x+1=0$, which has no real solutions.

• “f is a surjective function if the domain of f is $\mathbb{Z} \setminus {0}$ and codomain of f is $\mathbb{Q}$.”:

  • FALSE. For the same reason as above, we can never produce an output of 1.

Final Answer: The correct statements are:

• f is an injective function if the domain of f is $\mathbb{N}$.
• f is an injective function if the domain of f is $\mathbb{Z} \setminus {0}$.