Problem 87 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$k(x)=\sqrt{2 x-5}$$

Step-by-Step Solution

Answer

The domain of the function \(k(x)=\sqrt{2x-5}\) is \(x \in \left[\frac{5}{2}, \infty\right)\).

Step 1: Identify restrictions based on the function type

The restriction for a square root function like this is that the value inside the square root must be greater than or equal to zero, since we cannot take the square root of a negative number. So in this case, we need to find the values of \(x\) that satisfy \(2x-5\geq0\).

Step 2: Solve the inequality

Now, we'll solve the inequality \(2x-5\geq0\). To do this, we'll first isolate \(x\): \begin{align*} 2x - 5 &\geq 0 \\ 2x &\geq 5 \\ x &\geq \frac{5}{2} \end{align*}

Step 3: Write the domain

We found that the function is defined for values of \(x\) that are greater than or equal to \(\frac{5}{2}\). So, we can write the domain of the function \(k(x)\) using interval notation as: \[x \in \left[\frac{5}{2}, \infty\right)\]

Key Concepts

Understanding the Square Root Function

The square root function, often represented as \( \sqrt{x} \), is a fundamental mathematical construct where the output is the non-negative number that, when multiplied by itself, gives the input \( x \). This function is undefined for negative numbers because there is no real number which, when squared, results in a negative value. In the context of function domains, this means the expression inside the square root must be zero or positive.

• For the function \( k(x) = \sqrt{2x - 5} \), the term \( 2x - 5 \) needs to be at least 0.
• This restriction ensures that the function produces real and meaningful outputs.

As you become more familiar with square root functions, you will often begin by checking the domain constraints to know where your function is valid. You'll see how these constraints play a crucial role in understanding where your calculations make sense, guiding you through more complex math problems later on.

Solving Inequalities for Functions

Inequality solving is a vital skill when working with function domains, particularly for functions involving square roots. Let's take the inequality \( 2x - 5 \geq 0 \) as an example. Here, the process involves finding the set of \( x \)-values that satisfy this condition.

• First, we add 5 to both sides: \( 2x \geq 5 \).
• Next, we divide by 2 to isolate \( x \): \( x \geq \frac{5}{2} \).

By solving this inequality, we determine the permissible values for \( x \) that make the original condition true. The solution tells us where the function \( k(x) = \sqrt{2x - 5} \) is permitted to "operate" based on the defined criteria.

Expressing Domains with Interval Notation

Interval notation is a concise way of expressing ranges of values, which is particularly useful for stating function domains. It employs brackets to signify whether endpoints are included or not:

• Square brackets \( [ ] \) indicate the endpoint is included (equal sign in inequality).
• Parentheses \( ( ) \) show that the endpoint is not included.

For the function \( k(x) = \sqrt{2x - 5} \), where we found \( x \geq \frac{5}{2} \), the domain is written using interval notation as \( \left[ \frac{5}{2}, \infty \right) \):

• \( \left[ \frac{5}{2} \right. \) because \( x \) equals \( \frac{5}{2} \) is included.
• \( \left. \infty \right) \) since infinity itself is conceptual, not a concrete number and never included.

This notation quickly communicates the domain to others who are working with or reviewing the function, making it a neat and efficient method for conveying logical constraints.