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Quiz: Function Classifications

Test your understanding of symmetry, modulus functions, piecewise definitions, and function classification with these review questions.

1. Which of the following is an even function?

  1. \(f(x) = x^3 - x\)
  2. \(f(x) = 2x + 1\)
  3. \(f(x) = x^4 + 3x^2\)
  4. \(f(x) = \frac{1}{x}\)
Show Answer

The correct answer is C. An even function satisfies \(f(-x) = f(x)\). For option C: \(f(-x) = (-x)^4 + 3(-x)^2 = x^4 + 3x^2 = f(x)\). ✓ Polynomials with only even powers (including constants) are even. Options A and D are odd functions, and option B is neither (it has both an even power \(x^0 = 1\) and an odd power \(x^1\)).

Concept Tested: Even Function

2. A function \(f\) satisfies \(f(-x) = -f(x)\). This function is:

  1. Even
  2. Neither even nor odd
  3. Periodic
  4. Odd
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The correct answer is D. This is the defining algebraic property of an odd function: negating the input negates the output. Odd functions have rotational symmetry about the origin — rotate the graph \(180°\) around \((0, 0)\) and it looks the same. Classic examples include \(f(x) = x\), \(f(x) = x^3\), and \(f(x) = \frac{1}{x}\).

Concept Tested: Odd Function

3. Solve \(|3x - 1| = 8\).

  1. \(x = 3\) only
  2. \(x = 3\) or \(x = -\frac{7}{3}\)
  3. \(x = -3\) or \(x = \frac{7}{3}\)
  4. \(x = 3\) or \(x = \frac{7}{3}\)
Show Answer

The correct answer is B. Absolute value equations \(|A| = k\) split into \(A = k\) or \(A = -k\). So \(3x - 1 = 8\) gives \(3x = 9\) and \(x = 3\); or \(3x - 1 = -8\) gives \(3x = -7\) and \(x = -\frac{7}{3}\). Both solutions are valid. Check: \(|3(3) - 1| = |8| = 8\) ✓ and \(|3(-7/3) - 1| = |-8| = 8\) ✓.

Concept Tested: Absolute Value Equations

4. Describe how the graph of \(y = |f(x)|\) differs from the graph of \(y = f(x)\).

  1. Any portion below the \(x\)-axis is reflected above it
  2. The entire graph is shifted up
  3. The graph is mirrored about the \(y\)-axis
  4. Every \(y\)-value is squared
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The correct answer is A. The transformation \(y = |f(x)|\) takes the absolute value of the output. Wherever \(f(x)\) is positive, \(|f(x)|\) equals \(f(x)\) unchanged. Wherever \(f(x)\) is negative, \(|f(x)|\) is the positive mirror image — the graph is reflected up across the \(x\)-axis. Option C describes \(f(|x|)\), option D describes \([f(x)]^2\).

Concept Tested: Y Equals Mod F of X

5. A piecewise function is defined as \(f(x) = \begin{cases} x^2 & x < 1 \\ 3x - 1 & x \geq 1 \end{cases}\). What is \(f(1)\)?

  1. \(1\)
  2. \(0\)
  3. \(2\)
  4. \(4\)
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The correct answer is C. Check which interval \(x = 1\) belongs to. The first piece (\(x^2\)) applies when \(x < 1\) (strictly less than). The second piece (\(3x - 1\)) applies when \(x \geq 1\) — this includes \(1\). So \(f(1) = 3(1) - 1 = 2\). Always look carefully at whether the endpoint is included (\(\leq\), \(\geq\)) or excluded (\(<\), \(>\)) in each piece's condition.

Concept Tested: Piecewise Function

6. What is \(\lfloor -2.3 \rfloor\)?

  1. \(-2\)
  2. \(-3\)
  3. \(2\)
  4. \(3\)
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The correct answer is B. The greatest integer function (floor) returns the greatest integer less than or equal to \(x\). For negative numbers, this means rounding down (further from zero, not toward zero). The integers less than or equal to \(-2.3\) are \(\{..., -4, -3\}\) — notice \(-2\) is greater than \(-2.3\), so it doesn't qualify. The greatest of the valid integers is \(-3\).

Concept Tested: Greatest Integer Function

7. Classify \(f(x) = x^5 + 3x^3 - 2x\) as even, odd, or neither.

  1. Even
  2. Odd
  3. Neither
  4. Both even and odd
Show Answer

The correct answer is B. All powers of \(x\) are odd (\(x^5\), \(x^3\), \(x^1\)), so we expect an odd function. Verify algebraically: \(f(-x) = (-x)^5 + 3(-x)^3 - 2(-x) = -x^5 - 3x^3 + 2x = -(x^5 + 3x^3 - 2x) = -f(x)\). ✓ Only the zero function is both even and odd, so option D doesn't apply here.

Concept Tested: Symmetry Testing

8. The graph of \(y = f(|x|)\) is obtained from \(y = f(x)\) by:

  1. Taking the right half of \(f(x)\) (where \(x \geq 0\)) and mirroring it across the \(y\)-axis
  2. Reflecting the parts below the \(x\)-axis above it
  3. Shifting the graph left by 1 unit
  4. Squaring all \(y\)-values
Show Answer

The correct answer is A. The transformation \(f(|x|)\) replaces \(x\) with \(|x|\), which is always non-negative. So the function sees only \(|x|\), meaning inputs \(-2\) and \(2\) produce the same output. This makes the graph symmetric about the \(y\)-axis: take the right half of \(f\) (where \(x \geq 0\)) and reflect it to create the left half. The result is always an even function.

Concept Tested: Y Equals F of Mod X

9. The signum function \(\text{sgn}(x)\) returns which value when \(x = -5\)?

  1. \(-5\)
  2. \(5\)
  3. \(0\)
  4. \(-1\)
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The correct answer is D. The signum function returns the sign of its input: \(-1\) for negative inputs, \(0\) at zero, and \(+1\) for positive inputs. Since \(-5 < 0\), \(\text{sgn}(-5) = -1\). The signum function doesn't return the actual value or its magnitude — just a flag indicating the sign direction.

Concept Tested: Signum Function

10. Which function is NOT periodic?

  1. \(f(x) = \sin x\)
  2. \(f(x) = \cos x\)
  3. \(f(x) = x^2\)
  4. A function where \(f(x + 4) = f(x)\) for all \(x\)
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The correct answer is C. A periodic function satisfies \(f(x + T) = f(x)\) for some positive period \(T\). The parabola \(f(x) = x^2\) doesn't repeat — as \(x\) gets larger, the output grows without bound and never returns to previous values. Sine and cosine are classic periodic functions with period \(2\pi\), and option D describes a function with period \(4\) by definition.

Concept Tested: Periodic Function