Functions and Models MCQs in Calculus

By: Prof. Dr. Fazal Rehman Shamil | Last updated: February 12, 2025

Question 1:

\[
\text{Which of the following is the domain of the function } f(x) = \frac{1}{x-2}?
\]
\[
\text{(a) } (-\infty, 2), \quad \text{(b) } (-\infty, 2) \cup (2, \infty), \quad \text{(c) } [2, \infty), \quad \text{(d) } (-\infty, \infty)
\]
Answer: B

Step by Step Solution

Solution:

\[
\text{The function } f(x) = \frac{1}{x-2} \text{ is undefined when } x – 2 = 0.
\]
\[
\Rightarrow x = 2
\]
\[
\text{Thus, the domain excludes } x = 2, \text{ meaning the correct domain is } (-\infty, 2) \cup (2, \infty).
\]
\[
\boxed{(-\infty, 2) \cup (2, \infty)}
\]

Question 2:

\[
\text{If } f(x) = x^2 + 3x – 4, \text{ what is } f(2)?
\]
\[
\text{(a) } 6, \quad \text{(b) } 8, \quad \text{(c) } 10, \quad \text{(d) } 4
\]
Answer: B

Step by Step Solution

Solution:

\[
f(x) = x^2 + 3x – 4
\]
\[
f(2) = (2)^2 + 3(2) – 4
\]
\[
= 4 + 6 – 4
\]
\[
= 8
\]
\[
\boxed{8}
\]

Question 3:

\[
\text{Which of the following functions is an example of an exponential function?}
\]
\[
\text{(a) } f(x) = 2^x, \quad \text{(b) } f(x) = x^2, \quad \text{(c) } f(x) = x + 1, \quad \text{(d) } f(x) = \sin(x)
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{An exponential function has the form } f(x) = a^x, \text{ where } a > 0.
\]
\[
\text{The only function in the options that follows this form is } f(x) = 2^x.
\]
\[
\boxed{f(x) = 2^x}
\]

Question 4:

\[
\text{Which of the following is the inverse of } f(x) = 3x – 5?
\]
\[
\text{(a) } f^{-1}(x) = \frac{x+5}{3}, \quad \text{(b) } f^{-1}(x) = \frac{x-5}{3}, \quad \text{(c) } f^{-1}(x) = 3x + 5, \quad \text{(d) } f^{-1}(x) = \frac{x}{3} + 5
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{To find the inverse, replace } f(x) \text{ with } y:
\]
\[
y = 3x – 5
\]
\[
\text{Swap } x \text{ and } y:
\]
\[
x = 3y – 5
\]
\[
\text{Solve for } y:
\]
\[
3y = x + 5
\]
\[
y = \frac{x+5}{3}
\]
\[
\boxed{f^{-1}(x) = \frac{x+5}{3}}
\]

Question 5:

\[
\text{The graph of the function } f(x) = \sin(x) \text{ is:}
\]
\[
\text{(a) } \text{Always increasing}, \quad \text{(b) } \text{Always decreasing}, \quad \text{(c) } \text{Periodic}, \quad \text{(d) } \text{Linear}
\]
Answer: C

Step by Step Solution

Solution:

\[
\text{The function } \sin(x) \text{ repeats its values in a regular interval.}
\]
\[
\text{This means it is periodic with period } 2\pi.
\]
\[
\boxed{\text{Periodic}}
\]

Question 6:

\[
\text{Which of the following is the derivative of the function } f(x) = x^3 + 2x^2 – 5x?
\]
\[
\text{(a) } 3x^2 + 4x – 5, \quad \text{(b) } 3x^2 – 4x + 5, \quad \text{(c) } 3x^2 + 2x – 5, \quad \text{(d) } 2x^3 + 4x – 5
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{Using the power rule: } \frac{d}{dx} (x^n) = n x^{n-1}
\]
\[
\frac{d}{dx} (x^3) = 3x^2, \quad \frac{d}{dx} (2x^2) = 4x, \quad \frac{d}{dx} (-5x) = -5
\]
\[
\text{So, } f'(x) = 3x^2 + 4x – 5
\]
\[
\boxed{3x^2 + 4x – 5}
\]

Question 7:

\[
\text{The function } f(x) = \ln(x) \text{ is the inverse of which function?}
\]
\[
\text{(a) } e^x, \quad \text{(b) } 2^x, \quad \text{(c) } x^2, \quad \text{(d) } \sin(x)
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{The natural logarithm function } \ln(x) \text{ is the inverse of } e^x.
\]
\[
\text{That means } f^{-1}(x) = e^x \Rightarrow f(x) = \ln(x).
\]
\[
\boxed{e^x}
\]

Question 9:

\[
\text{Which of the following functions is even?}
\]
\[
\text{(a) } f(x) = x^3, \quad \text{(b) } f(x) = x^2, \quad \text{(c) } f(x) = \sin(x), \quad \text{(d) } f(x) = x + 1
\]
Answer: B

Step by Step Solution

Solution:

\[
\text{A function is even if } f(-x) = f(x) \text{ for all } x \text{ in the domain.}
\]

– **Option (a):** \( f(x) = x^3 \)
\[
f(-x) = (-x)^3 = -x^3 \neq f(x)
\]
\( \Rightarrow \) Not even.

– **Option (b):** \( f(x) = x^2 \)
\[
f(-x) = (-x)^2 = x^2 = f(x)
\]
\( \Rightarrow \) Even function.

– **Option (c):** \( f(x) = \sin(x) \)
\[
f(-x) = \sin(-x) = -\sin(x) \neq f(x)
\]
\( \Rightarrow \) Not even.

– **Option (d):** \( f(x) = x + 1 \)
\[
f(-x) = -x + 1 \neq f(x)
\]
\( \Rightarrow \) Not even.

Thus, the correct answer is:
\[
\boxed{f(x) = x^2}
\]

Question 10:

\[
\text{The function } f(x) = 2x – 3 \text{ represents:}
\]
\[
\text{(a) } A linear model, \quad \text{(b) } An exponential model, \quad \text{(c) } A quadratic model, \quad \text{(d) } A cubic model
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{A function is linear if it is in the form } f(x) = ax + b, \text{ where } a, b \text{ are constants}.
\]
\[
\text{The given function is } f(x) = 2x – 3.
\]

### Step 1: Check the highest power of \(x\)
– The highest power of \( x \) in \( f(x) = 2x – 3 \) is **1**.
– Linear functions always have the form \( ax + b \), where \( a \) is the slope.

### Step 2: Compare with given options
– **Option (a):** **Linear model** \( \checkmark \)
– **Option (b):** **Exponential model** \( (\text{Incorrect, as exponential functions have the form } a^x) \)
– **Option (c):** **Quadratic model** \( (\text{Incorrect, as quadratic functions have } x^2) \)
– **Option (d):** **Cubic model** \( (\text{Incorrect, as cubic functions have } x^3) \)

Since \( f(x) = 2x – 3 \) fits the form of a **linear function**, the correct answer is:
\[
\boxed{\text{A linear model}}
\]

Question 11:

\[
\text{Which of the following is the integral of } f(x) = 3x^2?
\]
\[
\text{(a) } x^3 + C, \quad \text{(b) } \frac{x^3}{3} + C, \quad \text{(c) } 3x^3 + C, \quad \text{(d) } 6x^2 + C
\]
Answer: A

Step by Step Solution

Solution:

\[
\text{The integral of a function } f(x) = ax^n \text{ follows the power rule:}
\]

\[
\int ax^n \,dx = \frac{a x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1.
\]

### Step 1: Identify \( a \) and \( n \)
– Given function: \( f(x) = 3x^2 \)
– Here, \( a = 3 \) and \( n = 2 \).

### Step 2: Apply the Power Rule
\[
\int 3x^2 \,dx = \frac{3x^{2+1}}{2+1} + C
\]
\[
= \frac{3x^3}{3} + C
\]

### Step 3: Simplify
\[
= x^3 + C
\]

### Step 4: Compare with Given Options
– **Option (a):** \( x^3 + C \) **(Correct)**
– **Option (b):** \( \frac{x^3}{3} + C \) **(Incorrect)**
– **Option (c):** \( 3x^3 + C \) **(Incorrect)**
– **Option (d):** \( 6x^2 + C \) **(Incorrect)**

Thus, the correct answer is:
\[
\boxed{x^3 + C}
\]

Question 12:

\[
\text{The function } f(x) = e^x \text{ is:}
\]
\[
\text{(a) } \text{Always increasing}, \quad \text{(b) } \text{Always decreasing}, \quad \text{(c) } \text{Periodic}, \quad \text{(d) } \text{Linear}
\]
Answer: A

Step by Step Solution

Solution:

### Step 1: Understand the Given Function
– The function given is:
\[
f(x) = e^x
\]
where \( e \) is Euler’s number (\(\approx 2.718\)).

### Step 2: Find the Derivative
– The derivative of \( f(x) \) determines whether it is increasing or decreasing:
\[
f'(x) = \frac{d}{dx} e^x = e^x
\]
– Since \( e^x > 0 \) for all real numbers \( x \), the function is **always increasing**.

### Step 3: Analyze Other Options
– **Option (b) Always decreasing:** Incorrect, because \( e^x \) is never decreasing.
– **Option (c) Periodic:** Incorrect, because a periodic function repeats values at regular intervals, but \( e^x \) does not.
– **Option (d) Linear:** Incorrect, because \( e^x \) is an exponential function, not a linear one.

### Conclusion:
Since the function is **always increasing**, the correct answer is:
\[
\boxed{\text{(a) Always increasing}}
\]

Question 13:

\[
\text{What is the derivative of the function } f(x) = \cos(x)?
\]
\[
\text{(a) } \sin(x), \quad \text{(b) } -\sin(x), \quad \text{(c) } -\cos(x), \quad \text{(d) } \cos(x)
\]
Answer: B

Step by Step Solution

Solution:

### Step 1: Understand the Given Function
– The function given is:
\[
f(x) = \cos(x)
\]

### Step 2: Differentiate the Function
– Using the standard derivative rule for cosine:
\[
\frac{d}{dx} \cos(x) = -\sin(x)
\]

### Step 3: Analyze the Options
– **Option (a) \( \sin(x) \)**: Incorrect, since the derivative of \( \cos(x) \) is **negative** sine.
– **Option (b) \( -\sin(x) \)**: Correct, as calculated above.
– **Option (c) \( -\cos(x) \)**: Incorrect, as the derivative of \( \cos(x) \) is not \( -\cos(x) \).
– **Option (d) \( \cos(x) \)**: Incorrect, as differentiation of cosine does not return cosine.

### Conclusion:
Since the derivative of \( f(x) = \cos(x) \) is **\( -\sin(x) \)**, the correct answer is:
\[
\boxed{\text{(b) } -\sin(x)}
\]

Question 14:

\[
\text{Which of the following represents the equation of a rational function?}
\]
\[
\text{(a) } f(x) = \frac{x^2 – 1}{x + 2}, \quad \text{(b) } f(x) = x^2, \quad \text{(c) } f(x) = x + 1, \quad \text{(d) } f(x) = e^x
\]
Answer: A

Step by Step Solution

Solution:

### Step 1: Definition of a Rational Function
– A **rational function** is a function that can be expressed as the **quotient of two polynomials**:
\[
f(x) = \frac{P(x)}{Q(x)}
\]
where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \neq 0 \).

### Step 2: Analyzing the Given Options
– **Option (a) \( f(x) = \frac{x^2 – 1}{x + 2} \)**:
– The numerator \( x^2 – 1 \) and the denominator \( x + 2 \) are both polynomials.
– **This is a rational function.**

– **Option (b) \( f(x) = x^2 \)**:
– This is a polynomial function, **not a rational function**.

– **Option (c) \( f(x) = x + 1 \)**:
– This is also a polynomial function, **not a rational function**.

– **Option (d) \( f(x) = e^x \)**:
– This is an **exponential function**, **not a rational function**.

### Conclusion:
Since **only option (a)** is in the form of a rational function, the correct answer is:
\[
\boxed{\text{(a) } f(x) = \frac{x^2 – 1}{x + 2}}
\]

Question 15:

\[
\text{The limit of } \frac{1}{x} \text{ as } x \to 0 \text{ is:}
\]
\[
\text{(a) } \infty, \quad \text{(b) } 0, \quad \text{(c) } -\infty, \quad \text{(d) } \text{Undefined}
\]
Answer: D

Step by Step Solution

Solution:

### Step 1: Understanding the Limit
– The function given is:
\[
f(x) = \frac{1}{x}
\]
– We need to determine the limit of \( f(x) \) as \( x \to 0 \).

### Step 2: Evaluating the Left-Hand and Right-Hand Limits
#### Case 1: \( x \to 0^+ \) (Approaching 0 from the Right)
– As \( x \) gets **closer to 0 from the positive side** (i.e., small positive values of \( x \)),
\[
\frac{1}{x} \to +\infty
\]
– Example: \( f(0.1) = 10, \quad f(0.01) = 100 \), etc.

#### Case 2: \( x \to 0^- \) (Approaching 0 from the Left)
– As \( x \) gets **closer to 0 from the negative side** (i.e., small negative values of \( x \)),
\[
\frac{1}{x} \to -\infty
\]
– Example: \( f(-0.1) = -10, \quad f(-0.01) = -100 \), etc.

### Step 3: Conclusion
– Since the **left-hand limit** is \( -\infty \) and the **right-hand limit** is \( +\infty \), the two limits **do not match**.
– Therefore, the **overall limit does not exist**.

### Final Answer:
\[
\boxed{\text{Undefined}}
\]

More MCQs on Calculus

  1. Functions and Models MCQs in Calculus
  2. Limits and Derivatives MCQs in Calculus
  3. Differentiation Rules MCQs in Calculus
  4. Applications of Differentiation MCQs in Calculus
  5. Integrals MCQs in Calculus
  6. Applications of Integration MCQs in Calculus
  7. Techniques of Integration MCQs in Calculus
  8. Differential Equations MCQs in Calculus
  9. Parametric Equations and Polar Coordinates MCQs in Calculus
  10. Infinite Sequences and Series MCQs in Calculus
  11. Vectors and the Geometry of Space MCQs in Calculus
  12. Vector Functions MCQs in Calculus
  13. Partial Derivatives MCQs in Calculus
  14. Multiple Integrals MCQs in Calculus
  15. Vector Calculus MCQs in Calculus
  16. Second-Order Differential Equations MCQs in Calculus

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