The Functions { G $}$ And { F $}$ Are Defined As Follows. For Each Function, Find The Domain By Showing Appropriate Work. Write Your Answer As An Interval Or Union Of Intervals.1. { G(x) = \frac{x^2}{x+6} $}$
The Functions g(x) and f(x): Finding the Domain
In mathematics, 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 all possible values of x that can be plugged into the function without causing any problems. In this article, we will explore the functions g(x) and f(x) and find their domains by showing appropriate work.
The function g(x) is defined as:
g(x) = \frac{x^2}{x+6}
To find the domain of g(x), we need to determine the values of x for which the function is defined. In other words, we need to find the values of x that do not cause any problems when plugged into the function.
Step 1: Identify the values that cause problems
The function g(x) has a denominator of x+6. If x+6 is equal to zero, then the function is undefined, because division by zero is not allowed. Therefore, we need to find the value of x that makes x+6 equal to zero.
x + 6 = 0
Subtracting 6 from both sides gives us:
x = -6
So, the value of x that causes a problem is x = -6.
Step 2: Write the domain as an interval or union of intervals
Since the function g(x) is undefined at x = -6, we need to exclude this value from the domain. Therefore, the domain of g(x) is all real numbers except -6. We can write this as a union of intervals:
Domain of g(x) = (-∞, -6) ∪ (-6, ∞)
The function f(x) is defined as:
f(x) = \frac{x^2}{x-2}
To find the domain of f(x), we need to determine the values of x for which the function is defined. In other words, we need to find the values of x that do not cause any problems when plugged into the function.
Step 1: Identify the values that cause problems
The function f(x) has a denominator of x-2. If x-2 is equal to zero, then the function is undefined, because division by zero is not allowed. Therefore, we need to find the value of x that makes x-2 equal to zero.
x - 2 = 0
Adding 2 to both sides gives us:
x = 2
So, the value of x that causes a problem is x = 2.
Step 2: Write the domain as an interval or union of intervals
Since the function f(x) is undefined at x = 2, we need to exclude this value from the domain. Therefore, the domain of f(x) is all real numbers except 2. We can write this as a union of intervals:
Domain of f(x) = (-∞, 2) ∪ (2, ∞)
In this article, we explored the functions g(x) and f(x) and found their domains by showing appropriate work. We identified the values of x that cause problems and wrote the domains as intervals or unions of intervals. The domain of g(x) is (-∞, -6) ∪ (-6, ∞), and the domain of f(x) is (-∞, 2) ∪ (2, ∞).
The Functions g(x) and f(x): A Q&A Guide
In our previous article, we explored the functions g(x) and f(x) and found their domains by showing appropriate work. In this article, we will answer some frequently asked questions about the functions g(x) and f(x).
Q: What is the domain of a function?
A: 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 all possible values of x that can be plugged into the function without causing any problems.
Q: How do I find the domain of a function?
A: To find the domain of a function, you need to identify the values of x that cause problems. These values are usually found by setting the denominator of the function equal to zero and solving for x. You then exclude these values from the domain.
Q: What is the difference between the domain and the range of a function?
A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values that the function can produce.
Q: Can a function have a domain that is a single value?
A: Yes, a function can have a domain that is a single value. For example, the function g(x) = x^2 has a domain of all real numbers except -6, but the function f(x) = x^2 has a domain of all real numbers.
Q: Can a function have a domain that is a union of intervals?
A: Yes, a function can have a domain that is a union of intervals. For example, the function g(x) = x^2 has a domain of (-∞, -6) ∪ (-6, ∞), and the function f(x) = x^2 has a domain of (-∞, 2) ∪ (2, ∞).
Q: How do I determine if a function is defined at a particular value of x?
A: To determine if a function is defined at a particular value of x, you need to plug the value of x into the function and see if it produces a valid output. If the function produces a valid output, then it is defined at that value of x.
Q: Can a function be defined at a value of x that makes the denominator equal to zero?
A: No, a function cannot be defined at a value of x that makes the denominator equal to zero. This is because division by zero is not allowed.
Q: What is the significance of the domain of a function?
A: The domain of a function is significant because it tells us the set of all possible input values for which the function is defined. This is important because it helps us to understand the behavior of the function and to make predictions about its output.
In this article, we answered some frequently asked questions about the functions g(x) and f(x). We discussed the domain of a function, how to find the domain of a function, and the difference between the domain and the range of a function. We also discussed how to determine if a function is defined at a particular value of x and the significance of the domain of a function.