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Quiz: Relations and Functions

Test your understanding of relations, functions, domain, range, and function notation with these review questions.

1. Which of the following relations is a function?

  1. \(\{(1, 2), (1, 3), (2, 4)\}\)
  2. \(\{(0, 5), (1, 5), (2, 6)\}\)
  3. \(\{(3, 1), (3, 2), (4, 5)\}\)
  4. \(\{(2, 7), (4, 8), (2, 9)\}\)
Show Answer

The correct answer is B. A function requires that every input be paired with exactly one output. In option B, the inputs \(0\), \(1\), \(2\) each appear once — note that \(0\) and \(1\) both mapping to \(5\) is fine, because different inputs sharing an output is allowed. Options A, C, and D all have an input that is paired with two different outputs, which violates the "one output per input" rule.

Concept Tested: Function

2. In the equation \(y = 3x - 2\), which variable is the independent variable?

  1. \(y\)
  2. \(3\)
  3. \(x\)
  4. \(-2\)
Show Answer

The correct answer is C. The independent variable is the input variable — the one you choose freely. In \(y = 3x - 2\), you decide what \(x\) is, and then \(y\) is determined from that choice. So \(x\) is independent and \(y\) is the dependent variable. The numbers \(3\) and \(-2\) are constants (specifically, \(3\) is a coefficient and \(-2\) is a constant term), not variables.

Concept Tested: Independent Variable

3. Given \(f(x) = x^2 - 4x + 1\), what is \(f(3)\)?

  1. \(-2\)
  2. \(6\)
  3. \(4\)
  4. \(10\)
Show Answer

The correct answer is A. Substitute \(x = 3\) into the function rule: \(f(3) = (3)^2 - 4(3) + 1 = 9 - 12 + 1 = -2\). A common mistake is to forget to square first or to mis-sign the middle term. Function notation \(f(3)\) simply means "evaluate \(f\) at the input \(3\)" — wherever \(x\) appears in the formula, replace it with \(3\).

Concept Tested: Function Notation

4. What is the domain of \(f(x) = \sqrt{x - 5}\)?

  1. \((-\infty, 5)\)
  2. \((5, \infty)\)
  3. \([5, \infty)\)
  4. \((-\infty, \infty)\)
Show Answer

The correct answer is C. For a square root to produce a real number, the expression under the radical must be non-negative: \(x - 5 \geq 0\), so \(x \geq 5\). Since \(x = 5\) gives \(\sqrt{0} = 0\) (a valid output), the endpoint is included, so we use a square bracket. The interval is \([5, \infty)\). Infinity always gets a round bracket.

Concept Tested: Domain Restrictions

5. Which interval notation represents "\(x\) is greater than \(-2\) and less than or equal to \(4\)"?

  1. \([-2, 4]\)
  2. \((-2, 4]\)
  3. \([-2, 4)\)
  4. \((-2, 4)\)
Show Answer

The correct answer is B. "Greater than \(-2\)" means \(x > -2\) (strict, so the endpoint is excluded — use a round bracket). "Less than or equal to \(4\)" means \(x \leq 4\) (non-strict, so the endpoint is included — use a square bracket). Combining gives \((-2, 4]\). Square brackets mean "include the endpoint," round brackets mean "exclude."

Concept Tested: Interval Notation

6. What are the Cartesian coordinates of a point located 4 units left of the origin and 3 units down?

  1. \((4, 3)\)
  2. \((-4, 3)\)
  3. \((4, -3)\)
  4. \((-4, -3)\)
Show Answer

The correct answer is D. In Cartesian coordinates \((x, y)\), moving left means a negative \(x\)-value, and moving down means a negative \(y\)-value. Four units left gives \(x = -4\), and three units down gives \(y = -3\). So the point is \((-4, -3)\). This point lies in Quadrant III, where both coordinates are negative.

Concept Tested: Cartesian Coordinates

7. Which function is the identity function?

  1. \(f(x) = 1\)
  2. \(f(x) = x\)
  3. \(f(x) = x^2\)
  4. \(f(x) = x + 1\)
Show Answer

The correct answer is B. The identity function returns exactly the input it receives: \(f(x) = x\). So \(f(3) = 3\), \(f(-7) = -7\), and so on. Option A is a constant function, option C is a quadratic, and option D shifts every input by \(1\). The identity function's graph is a straight line through the origin with gradient \(1\).

Concept Tested: Identity Function

8. A function \(g\) has domain \(\{-1, 0, 1, 2\}\) and is defined by \(g(x) = 2x + 3\). What is the range?

  1. \(\{1, 3, 5, 7\}\)
  2. \(\{-1, 0, 1, 2\}\)
  3. \(\{1, 5, 7\}\)
  4. \(\{3, 5, 7, 9\}\)
Show Answer

The correct answer is A. Evaluate \(g\) at each element of the domain: \(g(-1) = 2(-1) + 3 = 1\), \(g(0) = 3\), \(g(1) = 5\), \(g(2) = 7\). The range is the set of outputs: \(\{1, 3, 5, 7\}\). Remember that the range collects the outputs, not the inputs (option B lists the domain).

Concept Tested: Range

9. Which of the following describes a discrete function?

  1. Temperature as a continuous function of time
  2. The height of a ball as a function of time during free fall
  3. The number of students in a class as a function of the class period
  4. The distance traveled as a function of elapsed time
Show Answer

The correct answer is C. A discrete function is defined only at isolated points — typically whole-number inputs. You cannot have \(2.5\) students, so "number of students" only makes sense for integer values. The other options involve continuous quantities (time flows smoothly), and their graphs would be unbroken curves rather than separate dots.

Concept Tested: Discrete Function

10. What is the domain of \(f(x) = \frac{2}{x + 3}\)?

  1. \(\{x \mid x \in \mathbb{R}, x \neq 3\}\)
  2. \(\{x \mid x \in \mathbb{R}, x > -3\}\)
  3. \(\{x \mid x \in \mathbb{R}, x \neq 0\}\)
  4. \(\{x \mid x \in \mathbb{R}, x \neq -3\}\)
Show Answer

The correct answer is D. The function has a denominator \(x + 3\), which becomes zero when \(x = -3\). Since division by zero is undefined, this value must be excluded from the domain. Every other real number is a valid input. The answer in set builder notation is \(\{x \mid x \in \mathbb{R}, x \neq -3\}\), or equivalently \((-\infty, -3) \cup (-3, \infty)\) in interval notation.

Concept Tested: Domain Restrictions