Graphs and tangents of functions of one variable

Question 1: Graph of a Function (from file image_989e54.png)

The Question: Consider a function $f: \mathbb{R} \to \mathbb{R}$ defined as $f(x) = x^0 + 1$. Which of the following options represents the graph of the function?

• ${(x, x^2+1) \mid x \in \mathbb{R}}$
• ${(x, x^2) \mid x \in \mathbb{R}}$
• ${(x, 1) \mid x \in \mathbb{R}}$
• The graph of $f$ is a subset of $\mathbb{Z} \times \mathbb{R}$.
• The graph of $f$ is a subset of $\mathbb{R} \times \mathbb{R}$.

Core Concept: The Graph of a Function and the Zero Exponent Rule

• Graph of a Function: The graph of a function $f$ is the set of all ordered pairs $(x, f(x))$ where $x$ is an element of the domain.
• Zero Exponent Rule: For any non-zero real number $x$, $x^0 = 1$. In the context of functions from $\mathbb{R} \to \mathbb{R}$, it is a standard convention to define $0^0=1$ to ensure continuity.

Detailed Solution:

1. Simplify the function: Following the convention, we define $x^0=1$ for all $x \in \mathbb{R}$.
   • $f(x) = x^0 + 1 = 1 + 1 = 2$.
   • This means that for any input $x$, the output is always 2. This is a constant function.
2. Represent the graph as a set: The graph is the set of all points $(x, f(x))$, which is ${(x, 2) \mid x \in \mathbb{R}}$.
3. Evaluate the options:
   • The first three options describe different functions and are incorrect.
   • “The graph of $f$ is a subset of $\mathbb{Z} \times \mathbb{R}$.” This would mean the x-coordinates must be integers ($\mathbb{Z}$), but the domain is all real numbers ($\mathbb{R}$). This is FALSE.
   • “The graph of $f$ is a subset of $\mathbb{R} \times \mathbb{R}$.” A graph on the Cartesian plane is a set of ordered pairs $(x, y)$ where both $x$ and $y$ are real numbers. This is the definition of the Cartesian product $\mathbb{R} \times \mathbb{R}$. Any function graph from $\mathbb{R} \to \mathbb{R}$ must be a subset of this. This is TRUE.

Final Answer: The graph of $f$ is a subset of $\mathbb{R} \times \mathbb{R}$.

Question 2: Tangent Lines to a Curve (from file image_984c91.png)

The Question: Let $C$ be a curve in the following Figure M2W1AQ 6. Which of the following statements are correct about the curve $C$?

Core Concept: Tangent Lines A tangent line to a smooth curve at a given point is a straight line that “just touches” the curve at that point without crossing through it. It has the same direction (slope) as the curve at that exact point. A line that cuts through the curve is a secant line.

Detailed Solution:

• Line $l_1$: This line cuts through the parabola at two distinct points. It is a secant line, not a tangent line.
• Line $l_2$: This line touches the parabola at its lowest point (the vertex). It does not cross through the curve. This is the definition of a tangent line at the vertex. The statement "$l_2$ is the tangent line at some point to the curve C" is TRUE.
• Line $l_3$: This line does not intersect the curve at all. Therefore, it cannot be a tangent line. The statement "$l_3$ is not a tangent line at some point to the curve C" is also TRUE.

Final Answer: The correct statements are:

• $l_2$ is the tangent line at some point to the curve $C$.
• $l_3$ is not a tangent line at some point to the curve $C$.

Question 3: Tangent vs. Secant (from file image_984c35.png)

The Question: Let $C$ be a curve in the following Figure M2W1AQ 7. Which of the following statements are correct about the curve $C$?

Detailed Solution:

Let’s examine the line labeled $\ell$ (which the options call $l_1$) and its relationship to the curve at the specified points.

• The line $\ell$ clearly intersects the curve at two points, $Q$ and $S$. A line that intersects a curve at two points is a secant line.
• At point P: The line $\ell$ passes above the curve. It does not touch the curve at P. Therefore, it is not a tangent at P.
• At point Q: The line $\ell$ passes through the curve. It is not a tangent at Q.
• At point R: The line $\ell$ passes below the curve. It does not touch the curve at R. Therefore, it is not a tangent at R.
• At point S: The line $\ell$ passes through the curve. It is not a tangent at S.

Based on this, the statement "$l_1$ is not a tangent line to the curve $C$ at $S$" is TRUE. All other statements claiming it is a tangent at P, Q, or R are false.

Final Answer: $l_1$ is not a tangent line to the curve $C$ at $S$.

Question 4: Identifying a Tangent Line (from file image_984bf7.png)

The Question: Let $C$ be a curve and three straight lines $l_1, l_2$ and $l_3$ pass through the point $P$. Which of the following statements are correct about the curve $C$?

Detailed Solution:

We must identify which of the three lines has the properties of a tangent at point P.

• Line $l_3$: This line clearly cuts through the curve at point P. It is a secant line.
• Line $l_2$: This line also cuts through the curve at point P, just at a different angle. It is also a secant line.
• Line $l_1$: This line just grazes the curve at point P, matching its curvature at that one point without crossing to the other side. This is the visual definition of a tangent line.

Final Answer: $l_1$ is the tangent line to the curve $C$ at $P$.

Question 5: Representing a Quadratic Function (from file image_984b7f.png)

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

Core Concept: Quadratic Function A quadratic function is a function of the form $f(x) = ax^2 + bx + c$ where $a \neq 0$. Its graph is the set of all points $(x, y)$ such that $y = ax^2 + bx + c$.

Detailed Solution: Let’s examine the set notation for each option. The notation ${(x, y) \mid \text{condition}}$ represents a set of points.

• Option 1: ${(x, x^3 + 2x + 1) \mid x \in \mathbb{R}}$. This represents $y=x^3+2x+1$, which is a cubic function.
• Option 2: ${(x, x^2 + 4x + 5) \mid x \in \mathbb{R}}$. This represents $y=x^2+4x+5$, which has a highest power of 2. This is a quadratic function.
• Option 3: ${(x, x^4 + 5x^2 + 3) \mid x \in \mathbb{R}}$. This represents $y=x^4+5x^2+3$, which is a quartic (degree 4) function.
• Option 4: ${(x, 9x^2) \mid x \in \mathbb{R}}$. This represents $y=9x^2$, which has a highest power of 2. This is a quadratic function.

Final Answer:

• ${(x, x^2+4x+5) \mid x \in \mathbb{R}}$
• ${(x, 9x^2) \mid x \in \mathbb{R}}$

Question 6: Cardinality of a Function’s Graph (from file image_984b7f.png)

The Question: Let $f: A \to B$ be a function, where A and B are subsets of $\mathbb{R}$ and cardinalities of the sets A and B are 4 and 4, respectively. What is the cardinality of $\Gamma(f)$, the graph of $f$?

Core Concept: Cardinality of a Graph The graph of a function $f$ with domain $A$ is the set of all ordered pairs $(x, f(x))$ for every $x$ in $A$. By definition, a function assigns exactly one output for each input. Therefore, the number of ordered pairs in the graph is exactly equal to the number of elements in the domain.

Detailed Solution:

1. The domain of the function is the set $A$.
2. The cardinality of the domain is given as $|A| = 4$.
3. The graph of the function consists of one point for each element in the domain.
4. Therefore, the cardinality of the graph, $|\Gamma(f)|$, must be equal to the cardinality of the domain $A$.
5. $|\Gamma(f)| = |A| = 4$.

Final Answer: The cardinality of the graph of $f$ is 4.

Questions 7 & 8: Tangents on a Non-Smooth Curve (from file image_984897.png)

Core Concept: Differentiability and Tangents A tangent line can only be defined at a point where a curve is smooth. A tangent line does not exist at sharp corners or cusps because the direction of the curve is not uniquely defined at those points.

7) Which of the following option(s) is(are) true?

Detailed Solution: Let’s examine the curve $C$ at each point.

• At point P: There is a sharp corner. The curve changes direction abruptly. Therefore, there is no tangent at P.
• At point Q: There is another sharp corner. Therefore, there is no tangent at Q.
• At points R and S: The curve appears smooth at these points, so a tangent would exist.
• At point T: The line segment is straight, so the tangent is the line itself.

Evaluating the options:

• “At point P, there is no tangent.” -> TRUE.
• “At point Q, there is no tangent.” -> TRUE.
• “At point P, there are an infinite number of tangents.” -> FALSE. There are none.

Final Answer:

• At point P, there is no tangent.
• At point Q, there is no tangent.

8) Let A be the set points where the curve C has no tangents. Find a lower bound for the cardinality of the set A.

Detailed Solution:

1. From the previous question, we identified that the curve has no tangents at the sharp corners, which are points P and Q.
2. Therefore, the set A must contain at least these two points: ${P, Q} \subseteq A$.
3. This means the cardinality of A, $|A|$, must be at least 2.
4. A lower bound for the cardinality is a number that it is guaranteed to be greater than or equal to.

Final Answer: A lower bound for the cardinality of the set A is 2.

Question 9: Cardinality of a Graph (General Case) (from file image_98483b.png)

The Question: Let $f: A \to B$ be a function, where A and B are finite subsets of $\mathbb{R}$. Which of the following options are correct?

Detailed Solution:

As explained in Question 6, the graph of a function $f$ is the set of ordered pairs $(a, f(a))$ for every single element $a$ in the domain $A$.

• The number of these pairs is therefore exactly equal to the number of elements in $A$. So, $|\Gamma(f)| = |A|$.

Let’s evaluate the options:

• “The cardinality of $\Gamma(f)$… is same as the cardinality of $A \times B$.”: FALSE. The cardinality of the Cartesian product is $|A| \times |B|$. The cardinality of the graph is just $|A|$.
• “The cardinality of $\Gamma(f)$… is same as the cardinality of $A$.”: TRUE. This is the definition.
• “The cardinality of $\Gamma(f)$… is same as the cardinality of $B$.”: FALSE. This would only be true if the function were a bijection (both one-to-one and onto) and $|A|=|B|$. It is not true in general.
• “The cardinality of $\Gamma(f)$… is less than or equal to the cardinality of $A \times B$.”: TRUE. Since $|\Gamma(f)| = |A|$ and $|A \times B| = |A| \times |B|$, and we assume B is not empty ($|B| \ge 1$), then $|A| \le |A| \times |B|$. While true, the second option is more precise.

Final Answer: The most precise correct statement is “The cardinality of $\Gamma(f)$, the graph of $f$ is same as the cardinality of $A$.”