Domain and Range of Absolute Value, Rational, Exponential, and Logarithmic Parent Functions
Description/Explanation/Highlights
Video Description
This video explores the domain and range of absolute value, rational, exponential, and logarithmic parent functions.
Steps and Key Points to Remember
To find the domain and range of the absolute value, rational, exponential, and logarithmic 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 absolute value parent function, \(y=\left|x\right|\), 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 the absolute value sign will cause any negative number to be changed to positive and we used all real numbers for the domain, y will be all real numbers greater than or equal to 0. \(\mathbb{R}, y\geq0\).
- We could also look at the graph of \(y=\left|x\right|\) and see that x-values extend to negative infinity on the left to positive infinity on the right but the y-values extend from 0 at the bottom of the “V” 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 \(\{y|y\in\mathbb{R}, y\geq0\}\) and is read, “the set of all y’s such that y is an element of the real numbers where y is greater than or equal to 0.”
- In interval notation it is written as \((0,\infty)\) and is read from 0 to infinity including 0 or inclusive of 0.
- We can use the same process for the rational parent function, \(y=\frac{1}{x}\).
- 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 except for 0 which would cause division by 0 which is undefined, so the domain is \(\mathbb{R}, x\neq0\), also written as \(\{x|x\in\mathbb{R}, x\neq0\}\) in set notation or \((-\infty, 0)\cup(0,\infty)\) in interval notation.
- Since there is a 1 in the denominator, we will never get 0 for y, therefore the range is \(\mathbb{R}, y\neq0\), also written as \(\{y|y\in\mathbb{R}, y\neq0\}\) in set notation or\((-\infty, 0)\cup(0,\infty)\) in interval notation.
- The exponential parent function, \(y=b^x\), is commonly represent by \(y=2^x\) for growth and \(y=\frac{1}{2}^x\) for decay, has the same domain and range regardless of whether the value of y is between 0 and 1 (decay) or greater than 1 (growth).
- Both parent graphs have a horizontal asymptote at the x-axis and although all positive and negative can be used for x, y will approach 0 but never actually get there.
- The domain is therefore \(\mathbb{R}\), but the range is \(\mathbb{R}, y>0\).
- The domain is written as \(\{x|x\in\mathbb{R}\}\) in set notation or \((-\infty, \infty)\) in interval notation.
- The range is written as \(\{y|y\in\mathbb{R}, y>0\}\) in set notation or \((0,\infty)\) in interval notation.
- The parent function \(y=\log_2{x}\) is the inverse of the exponential parent and since x- and y-values are switched to find the inverse, the domain and range are also switched. We cannot take the log of negative numbers or 0, but when we take logs of positive numbers (including fractions) we get both negative and positive numbers, as well as zero.
- The domain is therefore \(\mathbb{R}, x>0\) and the range is \(\mathbb{R}\) and can be written in set or interval notation as before.
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 \(y=\left|x\right|\) absolute value parent is \(\mathbb{R}\). The range is also \(\mathbb{R}\).
- The domain of the \(y=\frac{1}{x}\) is \(\mathbb{R}\) except \(x\neq0\) and the range is all real numbers \(\mathbb{R}\) where \(y\neq0\).
- The domain and range of the \(y=b^x\) parent function for both growth (where \(b>1\)) and decay (where \(0<b<1)\) is \(\mathbb{R}\) for the domain and all real numbers where \(y>0\) for the range.
- The domain and range of \(y=\log_2{x}\) is all real numbers greater than 0 for the domain (\(\mathbb{R}, x>0\)) and all real numbers for the range (\(\mathbb{R}\)).
- 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 zero, write \(\{x|x\in\mathbb{R}, x>0\}\) in set notation or \((0, \infty)\) in interval notation.
Video Highlights
- 00:00 Introduction
- 00: \(y=\left|x\right|\) absolute value parent function domain & range
- 03:00 \(y=\frac{1}{x}\) rational parent function domain & range
- 05:29 \(y=2^x\) exponential growth parent function domain & range
- 06:55 \(y=(\frac{1}{2})^x\) exponential decay parent function domain & range
- 07:24 \(y=\log_2{(x)}\) logarithmic parent function domain & range
- 09:07 Conclusion
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