Question

Find the domain of the function. $$y=\sqrt{x+9}$$

Short answer

The domain of the function \(y = \sqrt{x + 9}\) is all real numbers x such that \(x \geq -9\).

Step-by-step solution

Understand the requirement for domain in a square root function

A square root function \(f(x) = \sqrt{x}\) is only defined for \(x \geq 0\). Therefore, in the function \(f(x) = \sqrt{x+9}\), the expression \(x+9\) must be greater than or equal to zero.

Set up the inequality

In order for the square root not to be undefined, the value inside the root must not be negative. Hence, set up the inequality as \(x+9 \geq 0\).

Solve the inequality

To solve for \(x\), we want to isolate it on one side of the inequality, so we subtract 9 from both sides: \(x \geq -9\).

Key concepts

The Essence of Algebra

Algebra is a foundational branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a powerful tool that allows us to express general mathematical relationships and solve problems where some quantities are unknown.

In the context of our exercise involving the function \(y = \sqrt{x+9}\), algebra comes into play when we need to determine the domain, or the set of all possible values of x that we can plug into the function without breaking any mathematical rules.

The problem requires us to understand inequalities—a critical aspect of algebra. By setting up an inequality based on the properties of a square root function, we use algebra to solve for x, thus obtaining the domain of the function. Without algebra, we wouldn't be able to express or solve such problems effectively.

Square Root Function Characteristics

The square root function is a fundamental concept in algebra represented by \(f(x) = \sqrt{x}\). This function is particularly special because it only takes non-negative inputs—meaning the number under the square root, also known as the radicand, must be greater than or equal to zero.

If we try to take the square root of a negative number, we would enter the realm of complex numbers, which is beyond the scope of this exercise.

Visualizing The Square Root Function

The graph of a square root function forms a curve starting from the origin (0,0) and rising gently towards the right. This visual representation helps us remember that as x increases, so does \(\sqrt{x}\), but it does so at a decreasing rate.

This characteristic helps us understand why the domain derived from the equation \(y = \sqrt{x+9}\) includes every number greater than or equal to -9 but no less, because those would create a negative radicand.

Navigating Inequalities

Inequalities are relations between two expressions that may not be equal, represented by symbols such as \(<, \leq, >, \)and \(\geq\). They are pivotal in defining the domain for functions like square root functions where certain inputs may not be permissible.

When dealing with the square root function in our exercise, we encounter an inequality that emerges from the fact that the radicand, in this case \(x + 9\), must be non-negative for the function to have real number outputs.

Solving Inequalities

Solving inequalities often involves the same techniques as solving equations—adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. However, multiplying or dividing by a negative number flips the inequality sign. In our example, we subtract 9 from both sides, which does not flip the sign, leading us to the solution \(x \geq -9\). This procedure secures the domain and ensures all function values remain in the realm of the real numbers.