Function Range Comparison Find G(x) With Same Range As F(x) = -2√(x-3) + 8

Hey everyone! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to determine their ranges. We've got a question on the table that involves finding a function with the same range as f(x) = -2√(x-3) + 8. This is a classic problem that tests our understanding of transformations and the behavior of square root functions. So, let's put on our thinking caps and break it down step by step.

Decoding the Original Function: f(x) = -2√(x-3) + 8

Before we even glance at the answer choices, the most crucial step is to thoroughly understand the original function, f(x) = -2√(x-3) + 8. Let's dissect it piece by piece:

• The Square Root (√): This is our base function. The square root function, in its simplest form (√x), only accepts non-negative inputs (x ≥ 0) and produces non-negative outputs. This is because we can only take the square root of a non-negative number and the principal square root is always non-negative. Therefore, the basic range of √x is [0, ∞).
• The (x-3) Inside the Square Root: This represents a horizontal shift. Subtracting 3 from x inside the square root shifts the entire graph 3 units to the right. This means the domain of our function is now x ≥ 3, as we can't take the square root of a negative number. The function is only defined for values of x greater than or equal to 3.
• The -2 Multiplier: This does two things. First, the negative sign reflects the graph across the x-axis. Instead of the graph opening upwards, it now opens downwards. Second, the 2 stretches the graph vertically by a factor of 2. So, the outputs are now twice as large in magnitude.
• The +8: This is a vertical shift. Adding 8 to the entire function shifts the graph 8 units upwards. This directly affects the range of the function.

Now, let's put it all together to determine the range of f(x). The basic square root function √x has a range of [0, ∞). The -2 multiplier reflects the graph and stretches it, changing the range to (-∞, 0]. Finally, the +8 shifts the entire range upwards by 8 units. Therefore, the range of f(x) = -2√(x-3) + 8 is (-∞, 8]. This means the function can output any value less than or equal to 8.

To really solidify this understanding, think about what happens as x gets larger and larger (but still greater than or equal to 3). The √(x-3) term will grow, and when multiplied by -2, it will become a large negative number. Adding 8 will only slightly offset this large negative value. This confirms that the function's output can be any value less than or equal to 8.

Analyzing the Answer Choices

Okay, guys, now that we've meticulously dissected the original function and nailed down its range, let's put our detective hats on and examine the answer choices. Our mission is to identify the function that shares the same range of (-∞, 8].

A. g(x) = √(x-3) - 8

Let's break down this function:

• √(x-3): As we discussed earlier, this is a square root function shifted 3 units to the right. Its basic range is [0, ∞).
• - 8: This shifts the graph 8 units downwards. So, the range of g(x) becomes [0, ∞) - 8, which is [-8, ∞).

This range is definitely different from (-∞, 8], so we can rule out option A.

B. g(x) = √(x-3) + 8

Let's analyze this one:

• √(x-3): Again, this is a square root function shifted 3 units to the right, with a basic range of [0, ∞).
• + 8: This shifts the graph 8 units upwards. So, the range of g(x) becomes [0, ∞) + 8, which is [8, ∞).

This range is also different from (-∞, 8]. This function only outputs values greater than or equal to 8, so option B is not the answer.

C. g(x) = -√(x+3) + 8

Time to investigate option C:

• √(x+3): This is a square root function shifted 3 units to the left (note the +3). Its basic range is still [0, ∞).
• -: The negative sign reflects the graph across the x-axis, changing the range to (-∞, 0].
• + 8: This shifts the graph 8 units upwards. So, the range of g(x) becomes (-∞, 0] + 8, which is (-∞, 8].

Eureka! This range matches the range of our original function, f(x). Option C is a strong contender.

D. g(x) = -√(x-3) - 8

Just to be absolutely sure, let's examine the final option:

• √(x-3): Square root function shifted 3 units to the right, range [0, ∞).
• -: Negative sign reflects the graph, range becomes (-∞, 0].
• - 8: Shifts the graph 8 units downwards. So, the range of g(x) becomes (-∞, 0] - 8, which is (-∞, -8].

This range is different from (-∞, 8], so we can eliminate option D.

The Verdict: Option C is the Winner!

After a thorough analysis of each answer choice, we've confidently identified option C, g(x) = -√(x+3) + 8, as the function with the same range as f(x) = -2√(x-3) + 8. Both functions have a range of (-∞, 8].

Key Takeaways and Why This Matters

This problem highlights several crucial concepts in understanding functions:

• Transformations: Recognizing how horizontal and vertical shifts, reflections, and stretches affect the graph and range of a function is essential. We saw how adding or subtracting values inside or outside the square root, as well as multiplying by a constant, dramatically changes the function's behavior.
• The Parent Function: Understanding the basic shape and range of the parent function (in this case, √x) is the foundation for analyzing transformations. Once you know the parent function, you can predict how the transformations will alter its range.
• Step-by-Step Analysis: Breaking down complex functions into smaller, manageable parts makes the analysis much easier. We systematically examined each component of the function to determine its impact on the range.

Why is understanding the range of a function so important? Well, the range tells us the set of all possible output values of a function. This is critical in many applications, such as:

• Modeling real-world phenomena: When we use functions to model real-world situations (like the height of a projectile or the growth of a population), the range tells us the possible values of the quantity we're modeling. For example, if we're modeling the height of a ball, the range will tell us the maximum height the ball can reach.
• Optimization problems: In optimization problems, we often want to find the maximum or minimum value of a function. Understanding the range can help us identify these extreme values.
• Calculus: The concept of range is fundamental in calculus, particularly when dealing with limits, continuity, and derivatives.

By mastering the techniques for determining the range of a function, you'll be well-equipped to tackle a wide range of mathematical problems and real-world applications.

Practice Makes Perfect

To really solidify your understanding, try working through similar problems. Here are a few ideas:

• Change the transformations: Try functions like f(x) = 3√(x+2) - 5 or g(x) = -0.5√(x-1) + 10.
• Use different parent functions: Explore the ranges of functions involving absolute values, quadratics, or cubics.
• Graph the functions: Use graphing software or a calculator to visualize the functions and their ranges. This can provide a valuable visual aid to your understanding.

Keep practicing, guys, and you'll become range-finding pros in no time!

Which function has the same range as the function f(x) = -2√(x-3) + 8? The options are: A. g(x) = √(x-3) - 8, B. g(x) = √(x-3) + 8, C. g(x) = -√(x+3) + 8, D. g(x) = -√(x-3) - 8.

Find the Function with Matching Range to f(x) = -2√(x-3) + 8