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How to Find the Range of a Function in A Level Maths?

Published in Function Range 3 mins read

To find the range of a function in A Level Maths, you'll need to determine all possible output values (y-values) that the function can produce. Here’s a step-by-step guide, incorporating the provided references:

Steps to Determine the Range of a Function

  1. Label the function: Let your function be represented as y = f(x).

  2. Express x as a function of y: Rearrange the equation y = f(x) to isolate x, expressing it in terms of y. This will give you x = f⁻¹(y) (though you don't necessarily need to explicitly write it as the inverse).

  3. Find all values of y where the inverse is defined: Determine the domain of f⁻¹(y). In other words, find all the y values for which the expression you found in step 2 is valid and produces a real value for x.

  4. Eliminate undefined y-values: Identify any values of y that would make the inverse function undefined. This could involve looking for:

    • Division by zero.
    • Square roots of negative numbers.
    • Logarithms of non-positive numbers.
    • Any other restrictions based on the specific function.

  5. Write the range: The range of f(x) consists of all the y values that remain after eliminating those that make x = f⁻¹(y) undefined. This can be expressed in various notations, such as:

    • Inequality notation: e.g., y > 2
    • Set notation: e.g., {y ∈ ℝ : y > 2}
    • Interval notation: e.g., (2, ∞)

Example

Let's find the range of the function f(x) = 1/(x - 3)

  1. y = 1/(x - 3)

  2. Solve for x:

    • y(x - 3) = 1
    • x - 3 = 1/y
    • x = 1/y + 3

  3. Examine x = 1/y + 3. The only value of y that makes this undefined is y = 0, as it results in division by zero.

  4. Therefore, the range is all real numbers except 0.

  5. Write the range: y ∈ ℝ, y ≠ 0. Alternatively, (-∞, 0) ∪ (0, ∞).

Important Considerations

  • Domain: Always consider the domain of the original function f(x). While the steps above focus on the inverse, sometimes the domain of f(x) restricts the possible output values (range).

  • Sketching: Sketching the graph of f(x) can be extremely helpful in visualizing the range.

  • Turning Points: For quadratic functions or other functions with turning points, identifying the turning point is crucial for determining the range.

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