Problem 20 - College Algebra Essentials

Question

Determine whether each equation defines \(y\) as a function of \(x .\) $$ y=-\sqrt{x+4} $$

Step-by-Step Solution

Answer

Yes, the given equation \(y=-\sqrt{x+4}\) defines \(y\) as a function of \(x\), but only for \(x \geq -4\).

Step 1: Analysis of given equation

We are given the equation \(y=-\sqrt{x+4}\). For this equation to define \(y\) as a function of \(x\), every \(x\) value must correspond to exactly one \(y\) value.

Step 2: Check for function conditions

In case of square root functions, they will always meet this condition if the expression under the square root is greater than or equal to zero (since the square root of a negative number is not a real number). So we need to check on which conditions the expression under the square root, \(x + 4\), is greater than or equal to zero.

Step 3: Determine the condition

The expression \(x + 4\) will be greater than or equal to zero when \(x \geq -4\). Therefore, for all \(x\) values greater than or equal to \(-4\), this equation gives exactly one \(y\) value, which means it defines \(y\) as a function of \(x\) for \(x \geq -4\).

Key Concepts

Understanding the Function of x

In algebra, when we talk about a 'function of x,' we are describing a relationship between two variables, usually called x and y. The idea is that the y values are dependent on the x values, meaning for every x, there is a corresponding y that can be calculated using a specific rule or equation.

For example, in the given equation from the textbook, \(y=-\sqrt{x+4}\), y is defined as the negative square root of \(x+4\). This means for any value of x you choose that satisfies certain conditions, you can find exactly one related y value by performing the operation described in the equation. The fundamental requirement for a relation to be a function is that each input (x-value) must result in one and only one output (y-value). So functions are all about unique pairings between x and y, which this equation satisfies under the right conditions.

Square Root Functions

Square root functions are a type of radical function that feature a square root sign. The general form of a square root function is \(f(x) = \sqrt{x}\) or \(y=\sqrt{x}\). These functions take an input x and give the output which is the square root of x. They are unique because they only produce real number outputs for non-negative x values, since the square root of a negative number isn't real in the set of real numbers.

The equation in our exercise, \(y = -\sqrt{x+4}\), is a variation with a transformation applied to the basic square root function. It reflects the graph of the basic square root function over the x-axis, since it includes a negative sign in front of the square root, and shifts it 4 units to the left, due to the +4 inside the square root. This change affects the domain of the function, which is an essential concept tied closely to square root functions.

Domain of a Function

The domain of a function is the complete set of possible x values for which the function is defined and will produce real y values. Understanding the domain is crucial because it tells us what x values we can safely plug into a function without getting undefined or non-real results.

In the case of the square root function \(y = -\sqrt{x+4}\), the function is only defined for x values that make the expression \(x+4\) inside the square root non-negative. Thus, we say the domain of \(y = -\sqrt{x+4}\) includes all real numbers \(x \) such that \(x \geq -4\). Any x less than \-4\ would result in trying to take the square root of a negative number, which isn't possible in the context of real numbers.

Understanding the domain of a function guides us on how we can graph the function and interpret its behavior across different values of x. For students, mastering the concept of a function's domain is foundational for grasping more complex mathematical relationships and solving a wide range of algebraic problems.