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IB DP Maths AA SL Study Notes

2.1.1 Domain and Range Basics

Methods to Determine Domain and Range

Algebraic Method

  • Domain: The domain of a function refers to the set of all possible input values (typically represented as x values) for which the function is defined.
    • To determine the domain, one must consider where the function might be undefined. For instance, denominators in a fraction cannot be zero, and the square root of a negative number is not a real number.
    • Example: For the function f(x) = 1/(x-2), the domain encompasses all real numbers except x = 2 because the denominator cannot equal zero.

Understanding the behaviour of rational functions further elucidates the concept of domains. Explore the basics of rational functions for a deeper understanding.

  • Range: The range of a function pertains to the set of all possible output values (typically represented as y values) that a function can produce based on its domain.
    • To determine the range, one must consider the maximum and minimum values the function can achieve.
    • Example: For the function f(x) = x2, the range is y >= 0 because a squared number is always non-negative.

Graphical Method

  • Domain: By observing the horizontal spread of the graph, one can determine the domain. The leftmost and rightmost points of the graph dictate the domain.
    • Example: For a parabola that opens upwards with its vertex at the origin, the domain is all real numbers since it extends indefinitely in both the left and right directions.

For a comprehensive understanding of how quadratic functions can influence the domain and range, consider studying quadratic functions.

  • Range: By observing the vertical spread of the graph, one can determine the range. The highest and lowest points of the graph dictate the range.
    • Example: For the aforementioned parabola, the range is y >= 0 because the graph lies entirely above the x-axis.

Learning about basic transformations can provide further insight into how the range of functions can be affected.

IB Maths Tutor Tip: Mastering domain and range enhances your problem-solving skills, enabling you to assess functions' limitations and possibilities in various mathematical and real-world contexts effectively.

Real-World Application

Understanding the domain and range is not just a theoretical exercise. In real-world scenarios, these concepts can help in determining feasible solutions. For instance, in a business context, the domain might represent the possible investments, while the range might represent the potential returns or losses.

Continuous vs. Discrete Functions

Continuous Functions

  • A function is termed continuous if its graph is unbroken. This means that one can draw the graph without lifting the pen off the paper.
  • Continuous functions have an uninterrupted domain and range.
  • Example: The function f(x) = x2 is continuous since its graph is a smooth curve without any breaks or holes.

Discrete Functions

  • A function is termed discrete if it is defined only for specific, distinct values within its domain.
  • Discrete functions often arise in scenarios where data can only assume specific, distinct values.
  • Example: The number of students in a maths class as a function of time is discrete because you cannot have a fraction of a student.

For those interested in how discrete functions manifest in real-world data, examining the introduction to sigma notation can be enlightening.

IB Tutor Advice: Practice sketching functions' graphs to visually determine their domain and range, aiding memory retention and comprehension for exams and enhancing your analytical skills in mathematics.

Example Questions

1. Question: Determine the domain and range of the function f(x) = sqrt(x).

  • Solution:
    • Domain: Since the square root of a negative number is not a real number, the domain is x >= 0.
    • Range: The smallest value of f(x) is 0 (when x = 0), and it increases as x increases. Therefore, the range is y >= 0.

2. Question: Is the function representing the height of a bouncing ball as a function of time continuous or discrete?

  • Solution: The function is continuous because the height of the ball changes smoothly with time, even when it bounces.

3. Question: Determine the domain and range of f(x) = sin(x) + 2.

  • Solution:
    • Domain: The domain is all real numbers.
    • Range: The range is 1 <= y <= 3 as the sine function oscillates between -1 and 1, and adding 2 shifts the range up.

Understanding the properties and equations of logarithms, such as in logarithmic equations, is crucial for mastering the determination of domains and ranges in more complex scenarios.

FAQ

All functions have a domain and a range. By definition, a function assigns every element in its domain to an element in its codomain. Therefore, a function must have a domain. Similarly, since there are values in the domain that are assigned to values in the codomain, there must be a range. However, it's possible for the domain or range to be very limited or even a single value. For instance, a constant function, where every input gives the same output, has a domain of all real numbers but a range that is just that constant value.

Piecewise functions are defined by different expressions or rules for different intervals of their domain. When determining the domain and range of a piecewise function, one must consider each piece or segment of the function separately. The domain will be the union of all the intervals where the function is defined, and the range will be the union of the outputs from each segment. It's essential to carefully analyse each piece, especially at the endpoints, to ensure that the entire domain and range are captured.

The concepts of domain and range are fundamental in real-world applications. For instance, in business, the domain might represent possible investments, while the range might represent potential returns or losses. In science, the domain could represent the set of possible inputs to an experiment, while the range represents the set of possible outcomes. Understanding the domain and range can help in determining feasible solutions, predicting outcomes, and making informed decisions in various fields.

In the real number system, taking the square root of a negative number is undefined. This is because no real number multiplied by itself will result in a negative number. However, in the field of complex numbers, the square root of a negative number is defined. The square root of -1 is represented by the imaginary unit, denoted as 'i'. So, while the square root of a negative number doesn't have a real value, it does have a value in the complex number system. For the purposes of most high school maths courses, unless specifically dealing with complex numbers, the square root of a negative number is considered undefined.

The domain of a function refers to the set of all possible input values (typically x-values) for which the function is defined. It's essentially the set from which values can be chosen to be input into the function. On the other hand, the codomain is a set that represents all possible output values that a function could potentially have, not necessarily the values it actually takes. In other words, while the range is the set of actual output values, the codomain is more about potential outputs. For example, if a function maps real numbers to real numbers, its codomain is all real numbers, even if the function doesn't produce all real numbers as outputs.

Practice Questions

Given the function f(x) = (x^2 + 3x - 4) / (x - 3.47), determine its domain and range.

The domain of a function is determined by the values of x for which the function is defined. In this case, the function is undefined when the denominator is zero. Thus, x - 3.47 = 0 gives x = 3.47. Therefore, the domain of f(x) is all real numbers except x = 3.47.

For the range, since the function is a rational function and there are no restrictions on the numerator, the range is all real numbers except for values that make the function undefined. In this case, the function is undefined at x = 3.47, but this does not restrict the range. Therefore, the range of f(x) is all real numbers.

Consider the function g(x) = sqrt(9.65 - x). Determine the domain of this function.

The domain of a function involving a square root is determined by the values of x for which the expression inside the square root is non-negative. For the function g(x), the expression inside the square root is 9.65 - x. This expression will be non-negative when 9.65 - x >= 0. Solving for x, we get x <= 9.65. Therefore, the domain of g(x) is all real numbers such that x <= 9.65.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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