Function Range: Which Includes -4?

by Andrew McMorgan 35 views

Hey Plastik Magazine readers! Today, we're diving into a fun little math problem that involves figuring out which function's range includes the number -4. We've got four options lined up, each with a square root function, and our mission is to determine which one can actually spit out -4 as a possible 'y' value. So, grab your thinking caps, and let's get started!

Understanding Range and Square Root Functions

Before we jump into the options, let's make sure we're all on the same page about what the range of a function is and how square root functions behave. The range of a function is essentially all the possible output values (y-values) that the function can produce. For square root functions, there are a couple of key things to remember.

First, the square root of a real number is only defined for non-negative numbers. In other words, you can't take the square root of a negative number (at least not in the realm of real numbers, which is what we're dealing with here). This means that the expression inside the square root (the radicand) must be greater than or equal to zero. This constraint will help us understand the domain of each function, which in turn affects its range.

Second, the square root of a non-negative number is always non-negative. The principal square root is always non-negative. The square root of 0 is 0, and the square root of any positive number is a positive number. This fact is crucial because it tells us that the basic square root function, y = √x, will always produce y-values that are zero or greater. Any transformations applied to this basic function will then shift or stretch this range.

Knowing these two key points, we can analyze each option to see if it's possible for the function to output -4.

Analyzing the Options

Let's examine each of the given functions to determine if -4 lies within their range. We'll consider the constraints imposed by the square root and how the constant term affects the possible output values.

A. y=x−5y = \sqrt{x} - 5

In this function, we have the basic square root function, √x, and then we're subtracting 5 from it. We know that √x will always produce values greater than or equal to 0. Therefore, the smallest possible value for √x is 0. If √x = 0, then y = 0 - 5 = -5. As x increases, √x increases, and y increases as well. So, the range of this function is y ≥ -5. Since -4 is greater than -5, -4 is included in the range. To confirm, we can set y = -4 and solve for x: -4 = √x - 5 => 1 = √x => x = 1. Since we found a valid x value (x=1) that produces y=-4, this function does include -4 in its range. So, option A looks promising!

B. y=x+5y = \sqrt{x} + 5

Here, we're adding 5 to the square root function. Again, √x will always be greater than or equal to 0. So, the smallest possible value for √x is 0. If √x = 0, then y = 0 + 5 = 5. As x increases, √x increases, and y increases as well. This means the range of this function is y ≥ 5. Since -4 is less than 5, -4 is not included in the range of this function. We can quickly eliminate this option.

C. y=x+5y = \sqrt{x + 5}

In this case, we're adding 5 inside the square root. This affects the domain of the function, not directly the range. The expression inside the square root, x + 5, must be greater than or equal to 0. This means x ≥ -5. The smallest possible value for x is -5, which gives us y = √( -5 + 5) = √0 = 0. As x increases from -5, the value of x + 5 increases, and therefore, the value of √ (x + 5) also increases. So, the range of this function is y ≥ 0. Since -4 is less than 0, -4 is not included in the range of this function. Another option we can rule out.

D. y=x−5y = \sqrt{x - 5}

Similar to option C, here we're subtracting 5 inside the square root. This affects the domain. The expression inside the square root, x - 5, must be greater than or equal to 0. This means x ≥ 5. The smallest possible value for x is 5, which gives us y = √(5 - 5) = √0 = 0. As x increases from 5, the value of x - 5 increases, and therefore, the value of √(x - 5) also increases. So, the range of this function is y ≥ 0. Since -4 is less than 0, -4 is not included in the range of this function. We can eliminate this option as well.

The Solution

After analyzing all the options, we found that only function A, y=x−5y = \sqrt{x} - 5, includes -4 in its range. We verified this by showing that we can find a valid x-value (x = 1) that results in y = -4. Therefore, the answer is A.

Key Takeaways

  • Understanding the range of a function is crucial for determining possible output values.
  • Square root functions have specific constraints: the radicand (the expression inside the square root) must be non-negative, and the square root itself is always non-negative.
  • Transformations of the basic square root function (addition, subtraction, etc.) affect the range by shifting it up or down.
  • Finding a valid x-value that produces a specific y-value confirms that the y-value is within the range of the function.

So there you have it, Plastik Magazine fam! We successfully navigated this function range problem. Remember to always consider the constraints and transformations of functions when determining their range. Keep practicing, and you'll become a math whiz in no time! Peace out!