Problem 88 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$r(k)=\sqrt{3 k+7}$$

Step-by-Step Solution

Answer

The domain of the function \(r(k)=\sqrt{3k+7}\) is \(k \geq -\frac{7}{3}\) or in interval notation, \([-\frac{7}{3},+\infty)\).

Step 1: Identify the critical points

The critical point for our function is the value of k that makes the expression inside the square root equal to zero. To find this critical point, we set the expression inside the square root equal to zero and solve for k: \(3k + 7 = 0\)

Step 2: Solve for k

Now, our goal is to isolate k. To do that, first, we'll subtract 7 from both sides: \(3k = -7\) Next, we'll divide both sides by 3: \(k = -\frac{7}{3}\)

Step 3: Determine the domain

For the function to have a real output, the expression inside the square root should be greater than or equal to zero. From step 1, we found that the critical point is \(k = -\frac{7}{3}\). This means that the square root will have a real output for all values of k greater than or equal to \(-\frac{7}{3}\). Therefore, the domain of the function r(k) is \(k \geq -\frac{7}{3}\) or in interval notation, \([-\frac{7}{3},+\infty)\).

Key Concepts

Square Root Function

The square root function is a fundamental function in algebra. It involves taking the square root of a number or an expression. The important thing to remember about square root functions is that they can only accept non-negative inputs if we want real numbers as outputs. This means the expression inside the square root, called the radicand, must be greater than or equal to zero. When handling square root functions, always identify what can be inside the square root so you don't end up with imaginary numbers.

Here's a quick example: given a function like \( f(x) = \sqrt{x-2} \), we need \( x-2 \geq 0 \), so \( x \geq 2 \). This tells us the function is defined for all \( x \) values starting from 2 and going to infinity.

Critical Points in Algebra

Critical points refer to points where the behavior of a function changes. In our context, it means finding where the expression inside the square root becomes zero. This is because, at this point, the transition happens from when the expression could potentially be negative to when it starts being non-negative.

• Set the radicand equal to zero
• Solve the equation to find the critical point

In the example \( r(k) = \sqrt{3k + 7} \), setting the radicand \( 3k + 7 \) equal to zero helps find the critical point, solving \( 3k + 7 = 0 \) gives us \( k = -\frac{7}{3} \). This critical point helps determine the boundary for the domain of the function.

Solving Inequalities

Solving inequalities is crucial when determining the domain of functions involving radicals or other conditions. The aim is to find all the values that satisfy the inequality.

• First, solve the related equation to find critical values.
• Use these values to test intervals on a number line.
• Determine where the inequality holds true.

For \( r(k) = \sqrt{3k + 7} \), after finding \( k = -\frac{7}{3} \), solve \( 3k + 7 \geq 0 \) to find \( k \geq -\frac{7}{3} \). This tells us that for any value \( k \) greater than or equal to \( -\frac{7}{3} \), the inequality holds true and the function is defined.

Interval Notation

Interval notation provides a mathematical shorthand to describe a set of numbers lying between two endpoints. It is commonly used for expressing domains.

• A square bracket \([ \text{or} ] \) denotes that an endpoint is included.
• A parenthesis \(( \text{or} ) \) denotes that an endpoint is not included.

For the domain of the function \( r(k) = \sqrt{3k + 7} \), we found \( k \geq -\frac{7}{3} \). In interval notation, this domain is written as \([ -\frac{7}{3}, +\infty)\), meaning all \( k \) values starting from \(-\frac{7}{3}\) (including it) to \( +\infty \). This notation makes it easy to visualize and communicate the range of possible \( k \) values succinctly.