When we talk about figuring out what kind of functions we have, knowing about domain and range is very important. Let’s simplify this!

What are Domain and Range?

• Domain: This is a fancy word that means all the possible input values (usually called $x$) for a function. For example, if we have the function $f(x) = \sqrt{x}$, the domain is $x \geq 0$. That’s because you can't take the square root of a negative number.

• Range: This refers to all the possible output values (usually called $y$) of a function. Using the same example, the range of $f(x) = \sqrt{x}$ is also $y \geq 0$, since the output will never be negative.

How Domain and Range Help Identify Function Types

1. Linear Functions: For a linear function like $f(x) = 2x + 3$, the domain and range are both all real numbers ($(-\infty, \infty)$). This means the function has no breaks and is smooth all the way through.

2. Quadratic Functions: In the case of a quadratic function like $g(x) = x^2$, the domain is all real numbers, but the range is $y \geq 0$. This means the graph has a U-shape that opens up.

3. Rational Functions: For a function like $h(x) = \frac{1}{x}$, the domain is $x \neq 0$ because you can’t divide by zero. The range is also $y \neq 0$. This shows that the graph has a vertical line that it won’t touch at $x = 0$ and doesn’t cross the x-axis.

4. Trigonometric Functions: For functions like $k(x) = \sin(x)$, the domain includes all real numbers, while the range is $-1 \leq y \leq 1$. This shows that these functions repeat in a regular pattern.

To sum it all up, looking at the domain and range helps us understand different types of functions. It also gives us clues about what the graphs of these functions will look like. Knowing these ideas is super helpful for understanding function graphs in Year 11 Math!