X^2 - 25/2 AND 37/K^2-K-30 I really need help with these two problems.?
•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
division by zero is not defined. What is 12/0? 0/0?
So, for a rational function, the domain excludes any values where the denominator is zero, because the domain is the set of values of x for which f(x) is defined.
For f(x) = (x^2 - 25)/2 the domain is all real numbers; there is no value of x which does not define a value for f(x).
In fact, the domain of all polynomial functions is the same: all real numbers.
For f(k) = 37/(k^2-k-30) we have to watch out for a zero denominator. Since
k^2-k-30 = (k-6)(k+5), the domain excludes the values -5 and 6.
Thank you Steve so vey much.You were a big help to me.
16t^2 – 1/64
To start, let's first explain the meaning of domain in mathematics. The domain of a function or expression is the set of all possible input values for which the function or expression is defined. In simpler terms, it refers to all the valid numbers or values that we can plug into an expression without encountering any issues.
Now, let's discuss why a denominator cannot be zero. In a rational expression, the denominator represents the part of the expression that cannot be equal to zero. This is because dividing by zero is undefined in mathematics. When the denominator of a fraction is zero, it indicates that the fraction is not well-defined or has no meaning. Therefore, to ensure that our expressions are properly defined, we must exclude any values for which the denominator would be zero.
Now let's find the domain for each of the given rational expressions:
1. x^2 - 25/2
Since this is a polynomial expression, there are no denominators involved. Hence, there are no values to exclude, and the domain is all real numbers. In set notation, we can write this as:
Domain: ℝ (set of all real numbers)
2. 37/(k^2 - k - 30)
In this rational expression, we have a denominator that involves a quadratic expression, k^2 - k - 30. To find the domain, we need to determine the values of k that would make the denominator equal to zero. So, we need to factor the quadratic expression: k^2 - k - 30 = (k - 6)(k + 5).
Setting each factor to zero gives us two possibilities: k - 6 = 0 or k + 5 = 0. Solving these equations, we find k = 6 and k = -5.
Therefore, the domain of the rational expression is all real numbers except 6 and -5. In set notation, this can be written as:
Domain: ℝ - {6, -5}
Incorporating the math vocabulary words:
1. Domain: The **domain** is the set of all possible input values for which a function or expression is defined.
2. Excluded value: An **excluded value** is a value that must be excluded from the domain to keep the expression defined or well-defined.
3. Set: A **set** is a collection of distinct objects or elements grouped together.
4. Factor: A **factor** is an algebraic expression that divides another expression evenly without leaving a remainder.
5. Real numbers: **Real numbers** include all rational and irrational numbers, representing quantities on the number line. The domain of our expressions includes real numbers except for specific excluded values.