Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$y=\sqrt{x+5}$$

Step-by-Step Solution

Answer

Yes, the equation \(y=\sqrt{x+5}\) represents \(y\) as a function of \(x\) because every value of \(x\) yields a unique value of \(y\).

Step 1: Determine the domain

\(y = \sqrt{x+5}\) is defined when \(x + 5 \geq 0\), i.e., \(x \geq -5\).

Step 2: Check the function criterion

For each \(x \geq -5\), the square root \(\sqrt{x+5}\) produces exactly one non-negative value. Therefore, each input \(x\) yields a unique output \(y\).

\(y = \sqrt{x+5}\) \(\textbf{does}\) represent \(y\) as a function of \(x\).

Key Concepts

Functions

Functions are fundamental concepts in algebra and mathematics as a whole. In simple terms, a function is a special relationship where each input has a unique output. Essentially, for a given input, you should always get the same output every time.

To determine if an equation is a function, you need to check if every input results in exactly one output. An easy way to visualize this is by using the "vertical line test" on a graph. If a vertical line crosses the graph more than once at any point, it's not a function. However, in many cases, especially when working with equations, this test can be done algebraically.

For the equation given in the exercise, \(y=\sqrt{x+5}\), each \(x\) value within a specified range gives one and only one \(y\) value. This aspect makes the equation a function because, in its domain, no \(x\) maps to multiple values of \(y\).

Domain and Range

When talking about functions, the domain and range are important concepts to grasp. The domain of a function is the complete set of possible values of the independent variable \(x\). In simpler terms, it is the list of "allowable" inputs into your function. For the equation \(y=\sqrt{x+5}\), the domain is \([-5, \infty)\). This is because \(x+5\) must not be negative inside the square root for real numbers; it needs to be zero or greater.

On the other hand, the range includes all possible outputs (or results) for the \(y\) values. For the equation \(y=\sqrt{x+5}\), since square roots of positive numbers or zero are always non-negative, \(y\) must be greater than or equal to zero. Thus, the range is \([0, \infty)\).

• The domain answers: "What \(x\) values can I use?"
• The range answers: "What \(y\) values can I get?"

Understanding these two concepts ensures you solve many mathematical problems correctly.

Square Roots

Square roots are a key mathematical concept and are involved when you need to find a number that multiplies by itself to achieve a given value. In simpler terms, if \(y\) is the square root of \(x\), then \(y \times y = x\).

The square root function is denoted by the radical symbol \(\sqrt{}\). It's crucial in our equation \(y=\sqrt{x+5}\). Here are a few important points to consider:

• The expression under the square root \(\sqrt{x+5}\) should never be negative in the realm of real numbers because a square root of a negative number becomes imaginary.
• This affects the domain since \(x \geq -5\) is needed to keep \(x+5 \geq 0\).
• Square roots produce non-negative numbers, resulting in a range starting at zero.

By mastering square roots, you can smoothly handle equations and problems involving this symbol, with clarity about what's possible and what's not.