When studying functions in Year 12 Mathematics, especially in the British AS-Level curriculum, it’s very important to understand the ideas of domain and range.

What is Domain?

The domain of a function includes all the possible values of the independent variable, usually called $x$.

To understand the domain better, you can look at the function's graph. The domain includes all the $x$-values that produce valid outputs. Here are some things to keep in mind:

1. Continuous Functions: Some functions don’t have any breaks or holes. If a function is continuous, its domain is often an interval. For example, the function $f(x) = x^2$ is valid for all real numbers, which we can write as $(-\infty, \infty)$.

2. Restrictions: Some functions cannot have certain values. For example, with $g(x) = \frac{1}{x}$, we cannot use $x = 0$. So, its domain is all real numbers except zero, written as $(-\infty, 0) \cup (0, \infty)$.

3. Square Roots and Logarithms: Functions that include square roots or logarithms have special rules. For instance, $h(x) = \sqrt{x}$ means $x$ must be greater than or equal to 0, making the domain $[0, \infty)$. For $j(x) = \log(x)$, $x$ must be greater than 0, so its domain is $(0, \infty)$.

Understanding Range

The range is all the possible values for the dependent variable, usually called $y$. Here’s how to figure it out:

1. Visual Inspection: You can also graph the function to see the range. For $f(x) = x^2$, this function only gives non-negative results. So, its range is $[0, \infty)$.

2. Behavior of Functions: For functions like $g(x) = x^3$, the outputs can be any real number, whether positive or negative. Therefore, its range is $(-\infty, \infty)$.

3. Trigonometric Functions: Sometimes, finding the range is a bit harder. For the sine function $k(x) = \sin(x)$, the highest value is 1 and the lowest is -1, so the range is $[-1, 1]$.

Using Intervals

Using intervals is a clear way to describe the domain and range. You can write:

• Domain:

  • Continuous: $(-\infty, \infty)$
  • With Restrictions: $(-\infty, 0) \cup (0, \infty)$

• Range:

  • Non-negative outputs: $[0, \infty)$
  • Unrestricted values: $(-\infty, \infty)$
  • Trigonometric: $[-1, 1]$

Practice Example

Let’s look at the function $f(x) = \frac{x^2 - 4}{x - 2}$.

1. Finding Domain: In this case, $x$ cannot equal 2 because that would make the bottom zero. So, the domain is $(-\infty, 2) \cup (2, \infty)$.

2. Finding Range: If we simplify the function and recognize there's a removable hole at $x = 2$, we can rewrite it as $f(x) = x + 2$ for all $x$ that isn’t 2. This means the outputs can be any real number, so the range is also $(-\infty, \infty)$.

In conclusion, using intervals to describe domain and range helps make everything clearer. As you learn more, remember to visualize these intervals so you can really understand these concepts!