Domain and Range of Linear, Quadratic, Square Root, Cubic, and Cube Root Parent Functions
Description/Explanation/Highlights
Video Description
This video explores the domain and range of linear, quadratic, square root, cubic, and cube root parent functions.
Steps and Key Points to Remember
To find the domain and range of the linear, quadratic, square root, cubic and cube root parent functions, follow these steps:
- Remember that in general, the domain is all of the possible x-values that can be used in a function and the range is all of the possible y-values that can result from inserting the x-values from the domain.
- The domain represents the possible inputs to the function, while the range represents the possible outputs.
- The domain is associated with the independent variable, while the range is associated with the dependent variable.
- To find the domain for the linear parent function, \(y=x\), we ask the question, “What are all of the x-values that we could put in for x?” In this case all numbers (positive, negative, zero, fractions, integers) could be substituted for x without creating an undefined y, therefore the domain for the linear parent is all real numbers (symbolized by \(\mathbb{R}\)).
- To find the range, do the same thing for the y-values. Ask, “What are the possible values that we could get for y by putting in the x-values in the domain?” In this case, since y=x and we used all real numbers for the domain, y will also be all real numbers \(\mathbb{R}\).
- We could also look at the graph of \(y=x\) and see that x-values extend to negative infinity on the left to positive infinity on the right and y-values extend from negative infinity at the bottom to positive infinity at the top.
- The domain could be written in set notation as\(\{x|x\in\mathbb{R}\}\) and is read, “the set of all x’s such that x is an element of the real numbers.”
- In interval notation it is written as \((-\infty,\infty)\) and is read, “from negative infinity to positive infinity.”
- The range written in set notation would be \(\{y|y\in\mathbb{R}\}\) and is read, “the set of all y’s such that y is an element of the real numbers.”
- In interval notation it is written and read the same as the domain, \((-\infty,\infty)\).
- We can use the same process for the quadratic parent function, \(y=x^2\).
- We ask, “What values of x can we put in that will get valid answers for y?” or where on the graph is x defined.
- Again, all real numbers satisfy the requirements for x, so the domain is \(\mathbb{R}\), also written as \(\{x|x\in\mathbb{R}\}\) in set notation or \((-\infty,\infty)\) in interval notation.
- However, unlike the linear function, when we put a negative into x, we will never get a negative number for y since squaring any number, negative or positive, always results in a positive y. Therefore the range is \(y\geq0\), also written as \(\{y|y\in\mathbb{R}, y\geq0\}\) in set notation or \([0, \infty)\) in interval notation (note the “[” on the left to denote that 0 is included).
- The square root parent, \(y=\sqrt{x}\), has a few more limitations.
- We cannot take the square root of a negative number so the domain of \(y=\sqrt{x}\) is \(x\geq0\), also written as \(\{x|x\in\mathbb{R}, x\geq0\}\) or \([0, \infty)\).
- Also, when we substitute values from the domain in for x, we will never get a negative y-value (assuming we use only the positive square root to keep it a function), therefore the range of \(y=\sqrt{x}\) is \(y\geq0\), also written as \(\{y|y\in\mathbb{R}, y\geq0\}\) or \([0, \infty)\).
- The cubic parent, \(y=x^3\) and all odd numbered exponential parents will have a domain and range of all real numbers just like the linear parent since all values of x can be used and y can be any number depending on the value from the domain used for x.
- The cube root parent, \(y=\sqrt[3]{x}\), and parents of all odd numbered roots, will also have all real numbers as its domain and range, just like the cubic function. This is not surprising since they are inverses of each other.
Here are some key points to keep in mind when identifying characteristics of parent functions and their graph.
- The domain for all parent functions is made up of all of the x-values that can substituted to get valid values for Y.
- The range consists all of the possible y-values that will result from substituting all the values in the domain for x.
- The domain represents the inputs into the function and is the independent variable (x).
- The range is represented by the dependent variables (y) and are the outputs of the function.
- The domain of the \(y=x\) linear parent is \(\mathbb{R}\). The range is also \(\mathbb{R}\).
- The domain of the \(y=x^2\) is also \(\mathbb{R}\) but the range is all real numbers where \(y\geq0\).
- The domain and range of the \(y=\sqrt{x}\) parent function is all real numbers where \(x\geq0\) and \(y\geq0\).
- The domain and range of \(y=x^3\) and its inverse, \(y=\sqrt[3]{x}\) are all \(\mathbb{R}\)
- The domain and range of any exponential or root parent function with an odd exponent or odd root will be \(\mathbb{R}\).
- Any domain or range can be expressed in set notation or interval notation. For example to write that the domain (x) is all the positive real numbers greater than or equal to zero, write \(\{x|x\in\mathbb{R}, x\geq0\}\) in set notation or \([0, \infty)\) in interval notation.
Video Highlights
- 00:00 Introduction
- 00:35 \(y=x\) linear parent function domain & range
- 03:45 \(y=x^2\) quadratic parent function domain & range
- 05:55 \(y=\sqrt{x}\) square root parent function domain & range
- 07:39 \(y=x^3\) cubic parent function domain & range
- 09:18 \(y=\sqrt[3]{x}\) cube root parent function domain & range
- 09:58 Conclusion
- To watch this video on YouTube in a new window with clickable highlights, click here