Read: Find Domain and Range From a Graph
Learning Objectives
- Define the domain of linear, quadratic, radical, and rational functions from graphs
![Line for f(x)=x+3](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/04/18232525/image046.gif)
Example
What are the domain and range of the real-valued function ?![Downward-opening parabola with vertex of 1, 4.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/04/18232527/image047.gif)
Answer: This is a quadratic function. There are no rational (divide by zero) or radical (negative number under a root) expressions, so there is nothing that will restrict the domain. Any real number can be used for x to get a meaningful output. Because the coefficient of is negative, it will open downward. With quadratic functions, remember that there is either a maximum (greatest) value, or a minimum (least) value. In this case, there is a maximum value. The vertex, or high point, is at (). From the graph, you can see that .
Answer
The domain is all real numbers, and the range is all real numbers f(x) such that . You can check that the vertex is indeed at (). Since a quadratic function has two mirror image halves, the line of reflection has to be in the middle of two points with the same y value. The vertex must lie on the line of reflection, because it’s the only point that does not have a mirror image! In the previous example, notice that when and when , the function value is . (You can verify this by evaluating and .) That is, both () and () are on the graph. The line of reflection here is , so the vertex must be at the point . Evaluating f(1) gives , so the vertex is at .Example
What is the domain and range of the real-valued function ?![Radical function stemming from negative 5, negative 2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/04/18232529/image048.gif)
Answer: This is a radical function. The domain of a radical function is any x value for which the radicand (the value under the radical sign) is not negative. That means , so . Since the square root must always be positive or , . That means .
Answer
The domain is all real numbers x where , and the range is all real numbers f(x) such that .Example
What is the domain of the real-valued function ?![Rational function](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/04/18232531/image049.gif)
Answer: This is a rational function. The domain of a rational function is restricted where the denominator is . In this case, is the denominator, and this is only when .