Different functions have different domains and ranges. Sometimes there are restrictions on domains and ranges.
The domain is the set of all input values. The range is the set of all output values. If we draw a graph with the x-coordinate as the horizontal values, and y-coordinate as the vertical values. Then the domain is where the function lies horizontally and the range is where the function lies vertically.
In the graph above, notice there are no values in the domain less than -5 on the left side of the graph. On the right side, the graph keeps going.
If we call this graph \(f(x)\), the domain is stated \(D: \{x\in \mathbb{R}|x \geq -5\}\). This means all x values (input values) can be any Real Number greater than or equal to -5. You can also state the domain as the following \(D:[-5, \infty) \). Notice a [ bracket was used before -5 to state that it is greater than or equal to. A circle bracket, ( ), is used for open ends like infinity, or when it is strictly greater than.
The range of the graph \(f(x)\) above, is stated as \(R: \{f(x)\in \mathbb{R}|f(x) \leq 5\}\). You can use \(y\) instead of \(f(x)\). This means that all \(y\) or \(f(x)\) values are less than or equal to 5. In the other notation, it is stated \(R:(-\infty, 5]\)
Knowing the properties and graphs of functions helps you find the domain and range.
For example, we know that \(f(x) = x^2\) is a parabola opening up with vertex at \((0,0)\).
The domain of \(f(x) \) is all the numbers, so \(D:\{x\in\mathbb{R}\}\).
The range is all the numbers greater than or equal to 0, so \(R:\{f(x)\in\mathbb{R}|f(x) \geq 0\}\).
Other quadratic functions will have similar domains and ranges.
For example, if \(f(x) = 3x^2-10\) is the same graph that has been stretched vertically by a factor of 3 and moved down 10 units.
The domain is still \(D:\{x\in\mathbb{R}\}\).
The range is now all the numbers greater than or equal to -10, since the vertex has moved down 10 units, so \(R:\{f(x)\in\mathbb{R}|f(x) \geq -10\}\).
So knowing the base function properties helps you determine domains and ranges after the graph has been transformed (dilated, moved. reflected).
Here are other common functions:
The root function, \(f(x) = \sqrt{x}\): \(D:\{x\in\mathbb{R}|x \geq 0\}\) and \(R:\{f(x)\in\mathbb{R}|f(x) \geq 0\}\)
The cubic function, \(f(x) = x^3\): \(D:\{x\in\mathbb{R}\}\) and \(R:\{f(x)\in\mathbb{R}\}\)
The reciprocal function \(f(x) = \frac{1}{x}\): \(D:\{x\in\mathbb{R}|x \neq 0\}\) and \(R:\{f(x)\in\mathbb{R}|x \neq 0\}\)