Problem 34 - Precalculus with Limits

Question

In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(y = -75\)

Step-by-Step Solution

Answer

Yes, the equation \(y = -75\) represents \(y\) as a function of \(x\).

Step 1: Analyze the given equation

The equation provided is \(y = -75\). In this equation, the value of \(y\) stays constant. No matter what the value of \(x\) may be, \(y\) will always be -75.

Step 2: Check the definition of a function

A function is a relation where every input (in our case \(x\)) is related to exactly one output (in our case \(y\)). Even though the value of \(y\) does not depend on \(x\) and is always -75, this still means that for each value of \(x\) there is exactly one value of \(y\). Thus, we see that the given equation meets the definition of a function.

Step 3: Conclusion

As we have tested the equation against the definition of a function and it met the criteria, we can conclude that the equation \(y = -75\) represents \(y\) as a function of \(x\).

Key Concepts

Constant Function

A constant function is one where the output value remains the same, no matter what the input is. In other words, if you choose any value for the variable (let's call it 'x'), the result after applying the function (the 'y' value) will always be the same.

For example, consider the equation \( y = -75 \). Here, regardless of the input value for \( x \), \( y \) will always be -75. It's like having a magic box that, no matter what you put into it, always gives you the same thing back, in this case, -75.

These types of functions are graphically represented by a horizontal line because the output never changes. So, for instance, if you were to plot this function on a coordinate plane, you would draw a straight line parallel to the x-axis at \( y = -75 \).

Definition of a Function

A function is a specific type of relation between sets of information. By definition, a function takes each element from a set (often referred to as the 'domain') and assigns it to exactly one element in another set (often called the 'range').

To say that in a more everyday way, a function is like a machine that has a certain rule. You give it an input (an element from the first set), and it gives you back an output (the element from the second set) that is determined by its rule. The important thing to remember is that for each input, there can only be one output.

Applying this to the task at hand, \( y = -75 \) meets the standards of a function because, for any value of \( x \) that we put into our 'machine,' we always get -75. There are no surprises here — one input leads to one and only one output.

Function of x

When we talk about a \( function \ of x \), we're discussing a rule that takes 'x' values (inputs) and gives us 'y' values (outputs). The letter 'x' is usually used to represent the variable in the domain, which are the values we can choose, and 'y' is the corresponding variable in the range, which are the values determined by the function.

For instance, in the equation \( y = -75 \), we say that 'y' is a function of 'x' because for every 'x' value we might consider, the rule tells us that 'y' is going to be -75. This relationship is true no matter what 'x' is; it could be 100, -20, or even a million. What's interesting about functions of 'x' is that they can describe all sorts of relationships, from simple ones like the constant function we've discussed to incredibly complex ones that can model the movement of planets or the growth of populations.