Problem 69 - College Algebra

Question

Find the domain of each function. $$ f(x)=\sqrt{24-2 x} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{24-2x}\) is all real numbers x such that \(x \leq 12\).

Step 1: Write down the inequality

To find the domain of the function, write down the inequality associated with it. Here, we have to solve the inequality \(24-2x \geq 0\).

Step 2: Solve the inequality

Solving the inequality, start by subtracting 24 from both sides of the inequality: \(-2x \geq -24\).\nThen, divide both the sides by -2. Remember, division by a negative number reverses the direction of the inequality. So, we get \(x \leq 12\).

Step 3: Write down the solution

The solution to the inequality \(x \leq 12\) is the set of all real numbers x such that x is less than or equal to 12. This is the domain of the given function.

Key Concepts

Inequalities in Algebra

Understanding inequalities is crucial in algebra—they are statements that describe a relationship where one side is not necessarily equal to the other. An inequality tells us that one side is larger (or smaller) than the other but does not indicate by how much. In our exercise, we're given the inequality associated with finding the domain of a function, which looks like this: \(24 - 2x \textgreater 0\).Inequalities come in various forms, like '>','<', '\(\textgreater\)' or '\(\textless\)' which signal strict inequalities, meaning there's no equality involved. Then there are '\(\textgreater\)' and '\(\textless\)' which are non-strict or inclusive inequalities, allowing for equality. The tricky part in algebra is remembering the rules, like flipping the inequality direction when multiplying or dividing by a negative number, as seen in our exercise.

• Write the inequality to show the constraint.
• Solve systematically, step by step.
• Remember to flip the inequality if you multiply or divide by a negative.

When dealing with inequalities in algebra, it's essential to work systematically to ensure that all implications of the inequality are considered. By following these rules and understanding the principles behind inequalities, students can confidently solve and graph such problems.

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that involves finding all possible solutions that satisfy the given inequality. It is similar to solving equations but with one critical difference: the direction of the inequality can change when we multiply or divide both sides by a negative number. This rule is crucial to remember as it can affect the entire solution set for the inequality.To illustrate, the function's domain in our exercise requires solving \(24 - 2x \geq 0\). Here are the steps:

1. Isolate the variable on one side by performing arithmetic operations.
2. Divide or multiply by negative numbers with caution, flipping the inequality symbol as needed.

Once you solve the inequality, the solution often includes a range of numbers, like \(x \leq 12\) in this case. This solution tells us that the domain of the function includes all real numbers less than or equal to 12. To improve comprehension, students should practice solving a variety of inequalities and use number lines to visualize the solution sets.
Regularly revisiting and practicing these rules enhance understanding and enable students to approach more complex inequalities with confidence.

Radical Functions

Radical functions, such as the one in our exercise, involve roots like square roots, cube roots, etc. These functions can have restrictions on their domains, as the radicand (the number under the root) must be greater than or equal to zero when dealing with even roots to produce real number outputs.In the function provided, \(f(x) = \sqrt{24-2x}\), the radicand is \(24-2x\). For the square root to be defined, the radicand must be non-negative. Thus, we write the inequality \(24 - 2x \geq 0\) to represent this condition and solve it accordingly. The domain of the function is the set of all 'x' values that satisfy this inequality.Here are important points to consider with radical functions:

• Even roots necessitate non-negative radicands for real functions.
• The domain involves solving inequalities derived from the radicand.
• Examining the radicand's behavior is key to determining the set of possible 'x' values.

By grasping the concept of radical functions and embodying the associated domain constraints, students can effectively sketch graphs of these functions and understand their behavior within their domains.