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Quiz: Foundations of Calculus

Test your understanding of functions and their properties with these review questions.

1. What is a function?

  1. An equation that can be graphed
  2. A rule that assigns exactly one output to each input
  3. Any mathematical expression with variables
  4. A relationship where inputs can have multiple outputs
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The correct answer is B. A function is a rule that assigns exactly one output to each input. This "one input, one output" property is what distinguishes functions from general relations. Option D describes a relation that is not a function.

Concept Tested: Function

2. The domain of a function is best described as:

  1. All possible output values
  2. The highest and lowest points on the graph
  3. All possible input values for which the function is defined
  4. The x-intercepts of the graph
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The correct answer is C. The domain is the set of all possible input values for which the function is defined. Option A describes the range, not the domain. Understanding domain restrictions is essential for calculus operations.

Concept Tested: Domain and Range

3. If f(x) = x² and g(x) = x + 3, what is (f ∘ g)(x)?

  1. x² + 3
  2. (x + 3)²
  3. x² + x + 3
  4. x² + 6x + 9
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The correct answer is B. The composition (f ∘ g)(x) means f(g(x)). First apply g to get x + 3, then apply f to get (x + 3)². Note that option D is the expanded form of option B, but B is the direct composition result.

Concept Tested: Composite Function

4. Which statement about inverse functions is correct?

  1. Every function has an inverse function
  2. The inverse of f(x) = x² is f⁻¹(x) = √x for all real numbers
  3. If f(a) = b, then f⁻¹(b) = a
  4. Inverse functions are always linear
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The correct answer is C. By definition, if f(a) = b, then the inverse function satisfies f⁻¹(b) = a. Option A is false because only one-to-one functions have inverses. Option B requires domain restriction. Option D is false—many inverse functions are nonlinear.

Concept Tested: Inverse Function

5. A piecewise function is defined as f(x) = {x² if x < 0; 2x if x ≥ 0}. What is f(−2)?

  1. −4
  2. 4
  3. −2
  4. 0
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The correct answer is B. Since −2 < 0, we use the first piece: f(−2) = (−2)² = 4. When evaluating piecewise functions, always check which condition the input satisfies before computing.

Concept Tested: Piecewise Function

6. The function f(x) = x³ is an example of:

  1. An even function because f(−x) = f(x)
  2. An odd function because f(−x) = −f(x)
  3. Neither even nor odd
  4. Both even and odd
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The correct answer is B. For f(x) = x³, we have f(−x) = (−x)³ = −x³ = −f(x), which is the definition of an odd function. Odd functions have origin symmetry, while even functions have y-axis symmetry.

Concept Tested: Even and Odd Functions

7. If f(x) = x², how does g(x) = (x − 3)² + 2 relate to f(x)?

  1. Shifted left 3 and down 2
  2. Shifted right 3 and up 2
  3. Shifted left 3 and up 2
  4. Shifted right 3 and down 2
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The correct answer is B. The transformation (x − 3) shifts the graph right 3 units (subtract inside = shift right), and the +2 outside shifts it up 2 units. Understanding transformations helps predict function behavior without graphing.

Concept Tested: Function Transformations

8. What is the end behavior of a polynomial function with a positive leading coefficient and odd degree?

  1. Both ends point upward
  2. Both ends point downward
  3. Left end down, right end up
  4. Left end up, right end down
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The correct answer is C. For odd-degree polynomials with positive leading coefficients, as x → −∞, f(x) → −∞, and as x → +∞, f(x) → +∞. This creates the "left down, right up" pattern typical of functions like f(x) = x³.

Concept Tested: Polynomial Function

9. What is one radian approximately equal to in degrees?

  1. 45°
  2. 57.3°
  3. 90°
  4. 180°
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The correct answer is B. Since π radians = 180°, one radian = 180°/π ≈ 57.3°. Radian measure is essential in calculus because derivative formulas for trigonometric functions only work correctly in radians.

Concept Tested: Radian Measure

10. Which trigonometric identity is known as the Pythagorean identity?

  1. sin(2x) = 2sin(x)cos(x)
  2. sin²(x) + cos²(x) = 1
  3. tan(x) = sin(x)/cos(x)
  4. cos(−x) = cos(x)
Show Answer

The correct answer is B. The identity sin²(x) + cos²(x) = 1 is called the Pythagorean identity because it comes from the Pythagorean theorem applied to the unit circle. This identity is used constantly in calculus for simplifying expressions and solving integrals.

Concept Tested: Trigonometric Identities