16t^2 – 1/64
Explain in your own words what the meaning of domain is. Also, explain why a denominator cannot be zero.
Find the domain for each of your two rational expressions.
Write the domain of each rational expression in set notation (as demonstrated in the example).
Do both of your rational expressions have excluded values in their domains? If yes, explain why they are to be excluded from the domains. If no, explain why no exclusions are necessary.
Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing your math work.
Domain
Excluded value
Set
Factor
Real numbers
a denominator cannot be zero because division by zero is undefined.
For this one, 64 is never zero.
The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of values that you can plug into the function to get a meaningful result.
In the given expression, 16t^2 - 1/64, there is no specific instruction about what t represents or what the function is defined for. So, we assume that t can be any real number for which the expression is meaningful. Therefore, the domain of this expression is the set of all real numbers, which can be represented in set notation as (-∞, ∞).
Now, let's talk about why a denominator cannot be zero. In mathematics, division by zero is undefined. When you divide a number by zero, it leads to mathematical inconsistency and contradiction. It is like trying to divide a pizza into 0 equal parts – it just doesn't make sense.
For rational expressions, where you have a fraction with a denominator, you need to ensure that the denominator is not equal to zero. Otherwise, the expression becomes undefined. In our case, there is no denominator involved in the given expression, so there is no risk of having a denominator equal to zero.
Hence, in this specific expression, there are no excluded values in the domain. All real numbers are valid inputs. Therefore, the domain is the set of all real numbers: (-∞, ∞).
Incorporating the math vocabulary words:
1. The domain of the expression is the set of all real numbers, which can be represented as (-∞, ∞).
2. There are no excluded values in the domain of the given expression.
3. The set notation used to represent the domain is (-∞, ∞).
4. The expression does not involve any factoring, as it is already simplified.
5. The domain consists of all real numbers, which are represented using set notation.