What is the domain of the following function: f(x) = 1/sqrt(x-4)?
Understand the Problem
The question asks for an example problem about the domain of a function. This means we need to provide a function and then ask what its domain is.
Answer
The domain of $f(x) = \frac{\sqrt{x-2}}{x-5}$ is $[2, 5) \cup (5, \infty)$.
Answer for screen readers
The domain of the function $f(x) = \frac{\sqrt{x-2}}{x-5}$ is $[2, 5) \cup (5, \infty)$.
Steps to Solve
- Define the Function
Let's define a function $f(x)$ as follows:
$$f(x) = \frac{\sqrt{x-2}}{x-5}$$
- Identify Potential Restrictions
The domain of a function is the set of all possible input values ($x$ values) for which the function is defined. In this case, we have two potential restrictions:
a. Square Root: The expression inside the square root must be greater than or equal to zero, because we cannot take the square root of a negative number and get a real number result. So, we need $x - 2 \ge 0$.
b. Division by Zero: The denominator cannot be zero, because division by zero is undefined. So, we need $x - 5 \ne 0$.
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Solve the Inequalities
a. For the square root restriction, $x - 2 \ge 0$, we add 2 to both sides to get $x \ge 2$.
b. For the division by zero restriction, $x - 5 \ne 0$, we add 5 to both sides to get $x \ne 5$.
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Combine the Restrictions
We need both conditions to be satisfied. So, $x$ must be greater than or equal to 2, but $x$ cannot be equal to 5.
- Express the Domain in Interval Notation
The domain can be expressed in interval notation as $[2, 5) \cup (5, \infty)$. This means all real numbers from 2 (inclusive) up to 5 (exclusive), and all real numbers greater than 5.
The domain of the function $f(x) = \frac{\sqrt{x-2}}{x-5}$ is $[2, 5) \cup (5, \infty)$.
More Information
The domain represents all possible $x$ values that can be plugged into the function $f(x)$ without resulting in an undefined expression. It's crucial to identify and address any restrictions imposed by square roots, division by zero, logarithms, or other operations that may limit the possible input values.
Tips
- Forgetting to consider both the square root and the division by zero restrictions.
- Incorrectly solving the inequality $x - 2 \ge 0$.
- Including 5 in the domain, even though it makes the denominator zero.
- Writing the interval notation incorrectly (e.g., using parentheses instead of brackets for the included endpoint 2).
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