Problem 20 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: Analyze the given equation

The equation given is \(y=-\sqrt{x}+4\). Here, for any value of \(x\), we will get a unique value of \(y\). This is the definition of a function.

Step 2: Evaluate the definition of function

A mathematical relationship is a function if each input corresponding to exactly one output. Thus for the given function \(y=-\sqrt{x}+4\), for any value of \(x\), we will get only one corresponding \(y\) value.

Step 3: Determination

Since there is exactly one output for each input, therefore \(y\) is a function of \(x\) in the equation \(y=-\sqrt{x}+4\).

Key Concepts

Defining Functions

Understanding what a function is remains a fundamental concept in precalculus, and it's crucial for moving forward with more complex mathematical analyses. At its core, a function is a specific type of relation where every input value is associated with exactly one output value. To put it simply, if you think of a function as a machine, whenever you insert a specific input (like a number), you'll always get a single, consistent output.

When we look at an expression such as \(y=-\frac{1}{2}x+4\), what's really being conveyed here is a rule. This rule tells us how to transform any ‘x’ (within the function's domain) into a corresponding ‘y’ (within its range). If you can replace ‘x’ with any number and get exactly one ‘y’ value every single time, you have yourself a function. In the context of our equation, for every possible \(x\), there's one and only one value of \(y\). This one-to-one correspondence is the defining trait of a function that sets it apart from more general relations.

Square Root Functions

Now, let's delve into the specifics of square root functions, which are a type of radical function characterized by the presence of a square root. The general form of a square root function is \(y = \sqrt{x}\), but it can also include additional transformations such as scaling, reflections, translations, and stretches. In our example, \(y=-\sqrt{x}+4\), we're dealing with a function that includes a square root transformation followed by a reflection across the \(x\)-axis (due to the negative sign) and a vertical shift upwards by 4 units.

It's worth noting that square root functions have an implied domain because you cannot take the square root of a negative number (in the set of real numbers). This restriction means that the domain of our function \(y=-\sqrt{x}+4\) only includes non-negative values of \(x\). Additionally, these functions are crucial for modeling scenarios where a quantity depends on the square of another, such as calculating the side of a square given its area.

Function Evaluation

Evaluating a function is akin to using a calculator: you plug in the input, press the equals sign, and receive the output. Function evaluation involves replacing the input variable in the function's formula with a specific value and calculating the result. When looking at the function \(y=-\sqrt{x}+4\), evaluating the function for a particular value of \(x\) means calculating the corresponding \(y\).

Let's say you want to evaluate the function for \(x = 9\). You would replace \(x\) with 9 in the equation, yielding \(y=-\sqrt{9}+4\), which simplifies to \(y=-3+4\); therefore, \(y=1\). This tells us that when \(x\) is 9, the output \(y\) is 1. This process is what allows us to create tables of values, plot graphs, or model real-world situations mathematically, providing an incredibly powerful tool for understanding all sorts of functions.