Problem 20 - College Algebra

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, as there is only one output value for each input value.

Step 1: Inspect the equation

Take a look at the given equation: \(y = \sqrt{x+4}\) . In this equation, for every value of \(x\) , there exists one and only one corresponding value of \(y\). This equation meets the requirement of a function, which is that each input \(x\) corresponds to exactly one output \(y\).

Step 2: Visualize if necessary

Though not required for this exercise, visualizing the function can often help. For this function, the graph would be a parabola that opens upward shifted to the left side by \(4\) units. This parabola does not violate the vertical line test, demonstrating this is a function.

Step 3: Coming to conclusion

After careful inspection and understanding of the equation, one can conclude that the equation \(y = \sqrt{x+4}\) does indeed represent \(y\) as a function of \(x\). Since for each and every value of \(x\) , there is a unique corresponding \(y\) value.

Key Concepts

Function Definition

Understanding the concept of a function is fundamental in algebra. A function is a special type of relation where every input, commonly denoted as x, is associated with exactly one output, often referred to as y. Think of it like a vending machine: for each selection you make (x), you get a specific item (y). The machine won’t give you more than one item for a single selection, this concept ensures that each input has a single, well-defined output.

When looking at an equation such as y = \(\sqrt{x+4}\), you have an example of a function since for each value of x that you put into the equation, you get out one specific value for y. The notation f(x) is also commonly used to denote a function, emphasizing the idea that f is a process that assigns to each input x exactly one output.

Vertical Line Test

The vertical line test is a quick visual way to determine if a curve is a graph of a function or not. If you can draw a vertical line, which means a line parallel to the y-axis, that intersects the curve more than once, then the curve is not the graph of a function.

This method works because if a vertical line crosses a curve at multiple points, then that vertical line is associated with multiple outputs for a single input, which breaks the rule of a function. For the equation y = \(\sqrt{x+4}\), a sketch would show that any vertical line would only cross the curve once. This confirms that the equation is, indeed, a function.

Square Root Functions

Square root functions are a type of radical function, specifically involving the square root. A general form of a square root function is y = \(\sqrt{x}\). However, they can also be modified with additional operations, such as addition and subtraction inside the square root or multiplying by a coefficient.

A key feature of square root functions is their domain: because you cannot take the square root of a negative number (in the set of real numbers), the domain is limited to non-negative numbers. The function y = \(\sqrt{x+4}\) has a domain of x >= -4, as any x value less than -4 would lead to finding the square root of a negative number.

Graphing Parabolas

Graphing parabolas is essential when studying quadratic functions. A parabola is the shape of the graph of any quadratic function, and it typically looks like a symmetrical curve that opens upwards or downwards. The equation of a quadratic function usually follows the form \(y = ax^2 + bx + c\).

For the function y = \(\sqrt{x+4}\), although this isn't a quadratic equation, the square root element does yield a parabola-like shape when graphed. This particular graph would start at the point (-4,0) and then curve upwards, because it reflects only the positive outputs of a quadratic function, constrained by the square root. The leftward shift by 4 units is due to the addition of 4 inside the square root, indicating that the function's graph is not centered on the y-axis as most basic parabolas.