The domain is defined as the set of $x$ values that we use to generate the graph of the quadratic curve. When we choose to work with any $x$ value, the domain is written as: $$ \begin{equation}\begin{aligned} x\in \mathbb{R}\\ \end{aligned}\end{equation} $$
This means that $x$ is chosen as any real number. It is also possible for us to work with a limited subset of $x$ values, for example: $$ \begin{equation}\begin{aligned} &x>0\\ -3\le &x\le 3\\ -4\le &x\le 7\\ \end{aligned}\end{equation} $$
Each of these are valid domains, that is, they are $x$ values that can be used to plot the quadratic curve.
The range of a function
The range is the set of $y$ values that are produced by passing the $x$ values belonging to the domain into the function. The function is essentially a machine that accepts an $x$ value as input and outputs a corresponding $y$ value: $$ \begin{equation}\begin{aligned} x\rightarrow f(x)\rightarrow y\\ \end{aligned}\end{equation} $$
The range of a function is usually written using set-builder notation. For example: $$ \begin{equation}\begin{aligned} &\{y:y<y_1\}\text{ or }\\ &\{y:f(x)<y_1\}\\ \end{aligned}\end{equation} $$
For a range with $2$ boundaries: $$ \begin{equation}\begin{aligned} &\{y:y_1<y<y_2\}\text{ or }\\ &\{y:y_1<f(x)<y_2\}\\ \end{aligned}\end{equation} $$
The range of a quadratic function
The turning point of a quadratic function will naturally have to be considered when finding its range:
- For $x\in \mathbb{R}$, a minimum curve will have a minimum value of $k$. Its maximum value will be $\infty$. The range is written as $$ \begin{equation}\begin{aligned} \{y:f(x)\ge k\}\\ \end{aligned}\end{equation} $$
- For $x\in \mathbb{R}$, a maximum curve will have a maximum value of $k$. Its minimum value will be $-\infty$. The range is written as $$ \begin{equation}\begin{aligned} \{y:f(x)\le k\}\\ \end{aligned}\end{equation} $$
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