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When we discuss a function of x, we're referring to a special relationship where each input value (labeled as x) is connected to exactly one output value. This connection is akin to a machine that, for every piece of input you feed it, gives back a distinct piece of output.
Consider the situation where you're given different numbers to plug into a mathematical formula, and as a result, you get a variety of outcomes. Each of these outcomes is the direct result of the particular number you started with. An example might be the formula for calculating the area of a square, where the length of a side (our x) determines the total area (our output).
However, things get a little quirky with the equation in our exercise, y=-75. Here, no matter what value x assumes, the output y remains a constant -75. This might seem unusual, but it is perfectly acceptable within the realm of functions. The key takeaway is that functions can come in various forms, some predictable and others, like the constant function here, surprising in their simplicity.

A horizontal line equation is one of the simplest equations you'll encounter in precalculus. It's an equation that describes a line that stretches left to right across the coordinate plane and does not tilt up or down. The general form of a horizontal line is y = b, where b is the y-coordinate through which the line runs.
For example, if we take the equation y = -75 from our textbook problem, this describes a horizontal line that lies 75 units below the x-axis. The important thing to remember about horizontal lines is that they do not change in y as x changes. Every point on this line has the same y value, which is what makes it horizontal. This property is exactly what makes our given equation a special kind of function – one that consistently outputs the same value no matter the input.

A function is a foundational concept in mathematics, especially in precalculus. It's so essential that we often compare it to a mathematical fingerprint: unique and specific in its assignment of outputs to inputs. Officially, a function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
This definition can be a mouthful, so let's break it down. Think of it as having a bag of socks (inputs) and a matching set of shoes (outputs). Each sock can go into only one shoe. If you find a sock that fits into two different shoes, we no longer call it a function.
The equation from our exercise, y = -75, checks off all the boxes in our function definition. Each value of x (our sock) pairs up with one and only one value of y (our shoe, which is -75 in this case). Thus, according to our textbook's step-by-step solution, the equation indeed represents y as a function of x. Even when the pairing seems overly simplistic or redundant, it still fits within our definition, showcasing the broad nature of functions.