Problem 89 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$g(t)=\sqrt{-t}$$

Step-by-Step Solution

Answer

The domain of the function \(g(t) = \sqrt{-t}\) is \(-\infty \leq t \leq 0\), or in interval notation, \((-∞, 0]\).

Step 1: Identify the condition for the square root function to be defined

The condition for the square root function to be defined is that the radicand must be greater than or equal to 0. In this case, the radicand is \(-t\).

Step 2: Find the domain by setting up an inequality

To determine the domain of the function, we need to find the values of \(t\) that make the radicand greater than or equal to 0. Thus, we must solve the following inequality: \[-t \geq 0\]

Step 3: Solve the inequality

Now, let's solve the inequality \(-t \geq 0\) for \(t\): \[t \leq 0\]

Step 4: Write the domain as an interval

Now that we have found the inequality \(t \leq 0\), we can write the domain of the function as an interval: \[\text{Domain}(g(t)) = (-\infty, 0]\] So, the domain of the function \(g(t) = \sqrt{-t}\) is \(-\infty \leq t \leq 0\), or in interval notation, \((-∞, 0]\).

Key Concepts

Inequality Solving

When finding the domain of a function like \(g(t) = \sqrt{-t}\), solving an inequality is essential. The expression under the square root, known as the "radicand," must satisfy specific conditions. For square roots to be defined with real numbers, the radicand should be non-negative. This means it must be greater than or equal to zero.
To determine this, set up the inequality with the radicand, in this case, \(-t \geq 0\). Solving this inequality requires simple steps:

• Reverse the sign of the inequality to change \(-t \geq 0\) into \(t \leq 0\).
• Understanding this solution means any value of \(t\) must be less than or equal to zero to satisfy the inequality. This process helps us find the specific range of values for which the original function is valid and defined.

Understanding how to set up and solve inequalities ensures the function can be evaluated without errors.

Square Root Functions

Square root functions, such as \(g(t) = \sqrt{-t}\), have unique properties that impact their domain. The critical element here is the value under the square root, known as the radicand, which must be non-negative for the function to produce real numbers.
In our example, \(-t\) plays this role, requiring us to ensure that it doesn’t produce a negative number.

• A square root function is only defined when the radicand is \(\geq 0\).
• This dictates that any transformations inside the square root affect domain limits, resulting in restrictions on the input values of \(t\). For \(\sqrt{-t}\), manipulating the radicand \(-t\) guides us to the inequality \(t \leq 0\).
• Functions like these can exhibit interesting shapes and behaviors on a graph, limited to defined domain intervals. This ensures that calculations remain within real numbers and avoid imaginary or undefined outputs.

By understanding these nuances, you can effectively identify the scope in which such functions operate.

Interval Notation

Interval notation serves as a concise way to express the domain of a function, which in this case stems from solving the inequality \(-t \geq 0\). After finding that \(t \leq 0\), we represent these values using interval notation.
This format visually and clearly communicates the set of all possible inputs.

• Here, \((-\infty, 0]\) denotes that \(t\) can take any value from negative infinity up to and including 0.
• The use of a parenthesis \(()\) indicates that the endpoint is not included, while a bracket \([]\) means it is included. So, \(0\) is part of the domain, while \(\infty\) is not bound.
• Interval notation is useful in mathematics for succinctly conveying complex ranges of values.

This approach to domains helps avoid lengthy explanations and provides clarity when working with functions across various mathematical contexts.