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About Functions & Graphs — Class 10 IB

Analyze domain, range, and transformations of polynomial, rational, and exponential functions. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 3). On this page you can practice 47 questions across three difficulty levels — 10 easy, 20 medium, and 17 hard — each with a visual step-by-step solution, plus a timed 34-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Functions & Graphs — solved examples for Class 10 IB

Example 1easy

Prisha is analyzing the function f(x) = 1/(x - 5). She states that the domain of this function is all real numbers. Which of the following statements correctly identifies and explains the error in Prisha's reasoning?
  1. A)Prisha is incorrect because the domain of a rational function excludes values that make the numerator zero. Here, x = 5 would make the numerator zero.
  2. B)Prisha is incorrect because the domain of a rational function excludes values that make the denominator zero. Here, x = 5 would make the denominator zero.
  3. C)Prisha is incorrect because the domain of a rational function excludes values that make the function negative. Here, x = 4 would make the function negative.
  4. D)Prisha is incorrect because the domain of a rational function excludes values that make the function undefined, but x = 5 makes the function defined as zero.

Step-by-step solution

  1. Step 1: Understand the definition of the domain for a rational function. The domain is the set of all possible input values (x) for which the function is defined.
  2. Step 2: Identify potential restrictions for rational functions. A rational function, which is a ratio of two polynomials, becomes undefined if its denominator is equal to zero.
  3. Step 3: Apply this rule to f(x) = 1/(x - 5). The denominator is (x - 5). Setting the denominator to zero gives x - 5 = 0, which means x = 5. Therefore, x = 5 must be excluded from the domain.
  4. Step 4: Conclude that Prisha's statement is incorrect because the domain excludes x = 5, making option B the correct explanation.

Answer: Prisha is incorrect because the domain of a rational function excludes values that make the denominator zero. Here, x = 5 would make the denominator zero.

Example 2medium

Consider the rational function f(x) = (2x - 1) / (x - 3). What is the domain of f(x)?
  1. A)x ≠ -3
  2. B)x ≠ 3
  3. C)x = 3
  4. D)All real numbers

Step-by-step solution

  1. The domain of a rational function includes all real numbers except for the values of x that make the denominator equal to zero.
  2. Set the denominator to zero: x - 3 = 0.
  3. Solve for x: x = 3.
  4. Therefore, the domain of f(x) is all real numbers such that x ≠ 3.

Answer: x ≠ 3

Example 3hard

Consider the function f(x) = sqrt(x^2 - 5x + 6) / (x - 4). Determine the domain of f(x).
  1. A)(-∞, 2] ∪ [3, ∞)
  2. B)(-∞, 2] ∪ [3, 4) ∪ (4, ∞)
  3. C)(-∞, 2) ∪ (3, ∞)
  4. D)(-∞, 4) ∪ (4, ∞)

Step-by-step solution

  1. Step 1: For the square root to be defined, the expression inside must be non-negative. So, x² - 5x + 6 ≥ 0. Factoring the quadratic gives (x - 2)(x - 3) ≥ 0. Analyzing the sign of this quadratic, it is non-negative when x ≤ 2 or x ≥ 3.
  2. Step 2: For the rational function to be defined, the denominator cannot be zero. So, x - 4 ≠ 0, which means x ≠ 4.
  3. Step 3: Combine both conditions. We need x ∈ (-∞, 2] ∪ [3, ∞) AND x ≠ 4. Therefore, the domain is (-∞, 2] ∪ [3, 4) ∪ (4, ∞).

Answer: (-∞, 2] ∪ [3, 4) ∪ (4, ∞)

Practice questions on Functions & Graphs

  1. Q1.easy

    Consider the exponential function f(x) = 3^x. Which of the following statements about its range is true?
    1. A)The range of f(x) is all real numbers (y ∈ ℝ).
    2. B)The range of f(x) is all real numbers greater than or equal to 0 (y ≥ 0).
    3. C)The range of f(x) is all real numbers greater than 0 (y > 0).
    4. D)The range of f(x) is all real numbers less than 0 (y < 0).
    Show answer

    Answer: The range of f(x) is all real numbers greater than 0 (y > 0).

    Hint: Think about the values an exponential function with a positive base can produce. Can it ever be negative or zero?

  2. Q2.easy

    The graph of a function y = f(x) is transformed to y = f(x + 2). Which of the following correctly describes the transformation?
    1. A)A horizontal shift of 2 units to the right.
    2. B)A horizontal shift of 2 units to the left.
    3. C)A vertical shift of 2 units up.
    4. D)A vertical shift of 2 units down.
    Show answer

    Answer: A horizontal shift of 2 units to the left.

    Hint: Remember that horizontal transformations inside the function behave in a way that might seem counter-intuitive to the sign.

  3. Q3.easy

    A graph of a function y = f(x) passes through the point (3, 7). If this function is reflected across the x-axis to create a new function g(x), which point must lie on the graph of g(x)?
    1. A)(-3, 7)
    2. B)(3, -7)
    3. C)(-3, -7)
    4. D)(7, 3)
    Show answer

    Answer: (3, -7)

    Hint: When a graph is reflected across the x-axis, how do the coordinates (x, y) change?

  4. Q4.medium

    A quadratic function is given by g(x) = -2(x - 3)² + k. If the maximum value of the function is 5, what is the value of k?
    1. A)3
    2. B)5
    3. C)-2
    4. D)-5
    Show answer

    Answer: 5

    Hint: For a quadratic function in vertex form a(x-h)²+k, the vertex is (h,k). The sign of 'a' determines if it's a maximum or minimum.

  5. Q5.medium

    The population P of a certain city (in thousands) t years after 2010 is modelled by the function P(t) = 50e^(0.03t). What was the initial population of the city in the year 2010?
    1. A)58 thousand
    2. B)55 thousand
    3. C)50 thousand
    4. D)52 thousand
    Show answer

    Answer: 50 thousand

    Hint: The initial population corresponds to the time when t = 0. Substitute t = 0 into the function to find the population at the starting year.

  6. Q6.medium

    A function's graph shows a continuous curve that passes through the origin, has a local maximum and a local minimum, and extends infinitely in both positive and negative x-directions. Which type of function is most likely represented by this graph?
    1. A)Linear function
    2. B)Quadratic function
    3. C)Cubic function
    4. D)Exponential function
    Show answer

    Answer: Cubic function

    Hint: Consider the typical shapes and characteristics of the graphs for each function type, especially the number of turning points (local maxima or minima).

  7. Q7.hard

    The graph of a function y = f(x) passes through the point (3, -2). If a new function g(x) is defined as g(x) = -2f(4x - 8) + 5, what are the coordinates of the corresponding point on the graph of g(x)?
    1. A)(1, 9)
    2. B)(5, 9)
    3. C)(1, 1)
    4. D)(5, 1)
    Show answer

    Answer: (5, 9)

    Hint: Work backwards from the transformation. First identify the input for f(x) and then apply the vertical transformations to the output of f(x).

  8. Q8.hard

    If f(x) is an even function, which of the following transformations applied to f(x) will always result in a function that is also even?
    1. A)h(x) = f(x + c) + d, where c=0
    2. B)h(x) = f(x + c) + d, where c ≠ 0
    3. C)h(x) = f(x) + d, where d ≠ 0
    4. D)h(x) = f(x) - x
    Show answer

    Answer: h(x) = f(x + c) + d, where c=0

    Hint: Recall the definition of an even function: f(-x) = f(x). Consider how each transformation affects the symmetry of the graph relative to the y-axis.

  9. Q9.hard

    Consider a piecewise function whose graph is described as follows:
    - For x < -2, the graph is a horizontal line segment at y = 3, with an open circle at x = -2.
    - For -2 ≤ x < 1, the graph is a line segment connecting (-2, 0) (closed circle) to (1, 3) (open circle).
    - For x = 1, there is an isolated point at (1, 1).
    - For x > 1, the graph is a parabola opening downwards with its vertex at (3, 5), passing through (1, 3) (open circle at x=1) and (5, 3) (open circle at x=5).
    What is the range of this function?
    1. A)(-∞, 5]
    2. B)[0, 5]
    3. C)[0, 3] ∪ (3, 5]
    4. D)[0, 3] ∪ {1} ∪ (3, 5]
    Show answer

    Answer: [0, 3] ∪ (3, 5]

    Hint: Carefully examine all segments of the graph and the y-values they cover. Pay close attention to open versus closed circles and the highest/lowest points reached.

These are 9 of the 47 questions available for Functions & Graphs. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.