Q1. Fill in the blank: A graph in the x-y plane represents a function if the graph passes the ______.

Background

Topic: Functions and their Graphs

This question tests your understanding of how to determine if a graph represents a function using a specific test.

Key Terms:

• Function: A relation where each input (x-value) has exactly one output (y-value).

• Vertical Line Test: A method to determine if a graph is a function.

Step-by-Step Guidance

1. Recall that a function assigns exactly one output to each input.

2. Think about what happens if you draw a vertical line through the graph at any x-value.

3. If the vertical line crosses the graph more than once, does this violate the definition of a function?

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Q2. True or False: There is a function on the real line, \( \mathbb{R} \), that does not have a limit anywhere.

Background

Topic: Limits and Pathological Functions

This question asks about the existence of functions with extreme discontinuity.

Key Terms:

• Limit: The value that a function approaches as the input approaches some value.

• Pathological Function: A function with unusual or extreme properties, often used as counterexamples.

Step-by-Step Guidance

1. Recall the definition of a limit at a point.

2. Think about whether you can construct or imagine a function that is so discontinuous that it fails to have a limit at every point.

3. Consider examples like the Dirichlet function or other highly oscillatory functions.

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Q3. A function \( f(x) \) with \( f(3) = -10 \) is continuous at \( x = 3 \) if, and only if, \( f(x) \) has a limit at \( x = 3 \) and the limit at \( x = 3 \) is ______.

Background

Topic: Continuity at a Point

This question tests your understanding of the formal definition of continuity at a point.

Key Terms and Formula:

• Continuity at a Point: \( f(x) \) is continuous at \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).

Step-by-Step Guidance

1. Recall the definition: \( f(x) \) is continuous at \( x = c \) if the limit exists and equals the function value at that point.

2. Given \( f(3) = -10 \), what must \( \lim_{x \to 3} f(x) \) be for continuity?

3. Look at the answer choices and select the one that matches the function value at \( x = 3 \).

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Q4. A function \( f(x) \) is continuous at \( x = c \) if, and only if, \( f(x) \) has a limit at \( x = c \) and \( \lim_{x \to c} f(x) = \) ______.

Background

Topic: Continuity at a Point

This is a restatement of the formal definition of continuity at a point.

Key Formula:

• \( f(x) \) is continuous at \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).

Step-by-Step Guidance

1. Recall the definition of continuity at a point.

2. Identify what the limit must equal for continuity at \( x = c \).

3. Fill in the blank with the correct expression.

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Q5. A function \( f(x) \) is continuous at a point \( c \) if, and only if, for every \( \varepsilon > 0 \) there is \( \delta > 0 \) such that whenever there is an \( x \) with \( |x - c| < \delta \), then ______.

Background

Topic: Epsilon-Delta Definition of Continuity

This question tests your understanding of the formal (epsilon-delta) definition of continuity.

Key Formula:

• \( f(x) \) is continuous at \( c \) if for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that whenever \( |x - c| < \delta \), then \( |f(x) - f(c)| < \varepsilon \).

Step-by-Step Guidance

1. Recall the epsilon-delta definition of continuity.

2. Identify what must be true about \( |f(x) - f(c)| \) when \( |x - c| < \delta \).

3. Fill in the blank with the correct inequality.

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Q6. Yes or No: Can a function \( f(x) \) have two limits at a point \( x = c \)?

Background

Topic: Uniqueness of Limits

This question tests your understanding of the uniqueness property of limits.

Key Terms:

• Limit: The value a function approaches as the input approaches a certain point.

• Uniqueness of Limits: If a limit exists at a point, it must be unique.

Step-by-Step Guidance

1. Recall the definition of a limit at a point.

2. Consider whether it is possible for a function to approach two different values as \( x \) approaches the same point from all directions.

3. Think about the formal definition and whether it allows for more than one limit at a point.

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Q7. A point \( x = c \) is said to be a root (or a zero) of a function \( f(x) \) if, and only if, \( f(c) = 0 \). Which theorem must we apply in order to claim that the function \( x^4 + x - 3 \) has a root in the interval [1, 2]?

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your understanding of the theorem that guarantees the existence of roots in a continuous function over an interval.

Key Terms and Theorem:

• Root (Zero): A value \( c \) such that \( f(c) = 0 \).

• Intermediate Value Theorem (IVT): If \( f \) is continuous on \( [a, b] \) and \( N \) is between \( f(a) \) and \( f(b) \), then there exists \( c \) in \( [a, b] \) such that \( f(c) = N \).

Step-by-Step Guidance

1. Recall the statement of the Intermediate Value Theorem.

2. Check if the function \( x^4 + x - 3 \) is continuous on \( [1, 2] \).

3. Determine if the function changes sign on the interval, which would indicate a root by the IVT.

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Q8. Given \( f(x) = \frac{x+1}{x} \) and \( g(x) = \frac{x+1}{x+2} \). Compute the following compositions and their domains:

Background

Topic: Function Composition and Domain

This question tests your ability to compute compositions of functions and determine their domains.

Key Terms and Formulas:

• Composition: \( (f \circ g)(x) = f(g(x)) \)

• Domain: The set of all input values for which the function is defined.

Step-by-Step Guidance (for part a: \( f \circ g(x) \))

1. Write out \( f(g(x)) \) by substituting \( g(x) \) into \( f(x) \).

2. Simplify the resulting expression as much as possible.

3. Determine the domain by considering where the denominator is zero in both \( g(x) \) and the composition.

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Q9. Sketch the graph of \( f(x) \) and compute its limits if possible, where:

\( f(x) = \begin{cases} x & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2 - x^2 & \text{if } 1 < x \leq 2 \\ x - 3 & \text{if } x > 2 \end{cases} \)

Background

Topic: Piecewise Functions and Limits

This question tests your ability to analyze piecewise functions and compute one-sided and two-sided limits.

Key Terms:

• Piecewise Function: A function defined by different expressions on different intervals.

• One-sided Limits: \( \lim_{x \to c^-} f(x) \) and \( \lim_{x \to c^+} f(x) \).

Step-by-Step Guidance (for part a: \( \lim_{x \to 1^-} f(x) \))

1. Identify which piece of the function applies as \( x \) approaches 1 from the left.

2. Substitute values close to 1 from the left into the appropriate expression.

3. Evaluate the limit using the left-hand piece.

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Q10. Is the function \( f(x) = |x| \) differentiable at \( c = 0 \)?

Background

Topic: Differentiability and Absolute Value Function

This question tests your understanding of differentiability at a point, especially for functions with sharp corners.

Key Terms and Formula:

• Differentiable: A function is differentiable at \( c \) if the derivative exists at that point.

• Derivative Definition: \( f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \)

Step-by-Step Guidance

1. Recall the definition of the derivative at a point.

2. Compute the left-hand and right-hand limits of the difference quotient at \( x = 0 \).

3. Compare the two one-sided limits to determine if they are equal.

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