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The domain of a function refers to all the possible input values (or "x" values) that a function can accept without causing any mathematical problems, such as division by zero or taking the square root of a negative number. For the function \( f(x) = -\sqrt{3 - 2x} \), we face a square root, which is only defined for non-negative numbers. This means the expression under the square root, \(3 - 2x\), must be greater than or equal to zero.

To find the domain, set up the inequality:

• \(3 - 2x \geq 0\).
• Solve for \(x\) to find \(2x \leq 3\) or \(x \leq \frac{3}{2}\).

Thus, the domain of the function \( f \) is all values of \(x\) that are less than or equal to \(\frac{3}{2}\). Understanding domains is crucial as they lay the groundwork for understanding how functions behave and which inputs are valid.

Square roots are a common component in mathematics, symbolized by the radical \(\sqrt{}\). The square root of a number \(a\) is a number \(x\) such that \(x^2 = a\). Importantly, square roots only produce non-negative results when working with real numbers. This is why our function \( f(x) = -\sqrt{3 - 2x} \) was constrained by the inequality \(3 - 2x \geq 0\).

Here’s what you need to remember about square roots:

• The expression inside the square root (called the radicand) must be non-negative in real number solutions.
• Squaring the square root cancels it out, allowing you to solve for unknowns if you isolate it appropriately.

We used this when we squared both sides in our step-by-step solution to eliminate the square root and solve for \(x\). Mastering square roots helps in solving equations and understanding shapes like circles and spheres in geometry.

Inequalities describe a relationship where one quantity is bigger or smaller than another. They are often depicted with signs such as \(>\), \(<\), \(\geq\), and \(\leq\). In our function \( f(x) = -\sqrt{3-2x} \), an inequality shows up when determining the domain. Calculating \(3 - 2x \geq 0\) tells us for which values of \(x\) the function is defined.

Handling inequalities involves a few key steps:

• Solve similarly to equations: isolate the variable on one side by performing valid operations on both sides.
• Careful with multiplication or division: If you multiply or divide both sides by a negative number, flip the inequality sign.
• Graphing: Visualize inequalities on a number line to better understand the solution set.

Understanding inequalities is essential for determining valid input values and understanding constraints within functions and real-world problems.

The range of a function includes all the possible output values (or "y" values) which come from the function's input domain. It's essentially the set of all possible end results a function can produce with its defined inputs. For \(f(x) = -\sqrt{3 - 2x}\), the outputs must be non-positive since the square root itself is non-negative, but it is multiplied by \(-1\).

To determine the range of \( f \), consider:

• The smallest possible value for \(y\): When the expression inside the square root is its maximum (at \(x = -\infty\)), \(y\) approaches 0.
• The largest possible value for \(y\): When \(x = \frac{3}{2}\), the expression inside is 0, making \(y = -\sqrt{0} = 0\).

Thus, the range of the function is \(-\sqrt{3} \leq y \leq 0\). Understanding a function's range is crucial for identifying potential outputs and analyzing a function's behavior.