Translated F(x)=|x|: What Happens To The Range?

Hey math enthusiasts and welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of function transformations, specifically focusing on the humble absolute value function, $f(x)=|x|$. You know, the V-shaped graph that's super important in calculus and beyond? We're going to take this function, give it a little nudge – a 6-unit shift to the right and a 2-unit hop upwards – and then figure out what happens to its range. This is crucial stuff, guys, because understanding how transformations affect a function's range is key to unlocking more complex mathematical concepts. We'll be breaking down the original function's properties, dissecting the translation process, and ultimately pinpointing the true statement about the range of both the original and the transformed function. So, grab your calculators, maybe a nice cup of coffee, and let's get this mathematical party started!

Understanding the Original Function: $f(x) = |x|$

Before we start messing around with translations, let's get reacquainted with our starting point: the function $f(x) = |x|$. This function is incredibly simple yet profoundly important. The absolute value of a number is its distance from zero on the number line, meaning it's always non-negative. So, if you input a positive number, you get that number back. If you input a negative number, you get its positive counterpart. For instance, $|5| = 5$ and $|-5| = 5$. When we graph this function, we get that iconic V-shape, with the vertex sitting right at the origin $(0,0)$. The domain of $f(x)=|x|$ is all real numbers, because you can plug in any real number and get a valid output. But what about the range? The range refers to the set of all possible output values (y-values) the function can produce. Since the absolute value of any real number is always zero or positive, the lowest output value this function can ever give us is 0. It can go up to positive infinity. Therefore, the range of $f(x)=|x|$ is all real numbers greater than or equal to 0. We can write this mathematically as ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 0}$ or in interval notation as $[0, ext{∞})$. It's super important to nail this down because it's our baseline for comparison. This range is fundamental to understanding how transformations will, or won't, affect our function's outputs. Keep this V-shape and its range firmly in your minds as we move on to the next step, where things get a bit more dynamic.

The Translation Transformation: Shifting Gears

Now, let's talk about the translation we're applying to our original function $f(x)=|x|$. Translations are one of the most basic yet powerful types of function transformations. They essentially involve shifting the entire graph of a function horizontally, vertically, or both, without changing its shape or orientation. In our case, we're performing a double translation: a horizontal shift of 6 units to the right and a vertical shift of 2 units up. Let's break down how these shifts affect the function's equation. When we shift a function $f(x)$ horizontally to the right by $h$ units, we replace $x$ with $(x-h)$ in the function's definition. So, for a shift of 6 units to the right, we'll be replacing $x$ with $(x-6)$. This gives us $|x-6|$. Next, when we shift a function vertically up by $k$ units, we simply add $k$ to the entire function. Here, we're shifting up by 2 units, so we'll add 2. Combining these two transformations, our new function, let's call it $g(x)$, will be defined as $g(x) = |x-6| + 2$. This new function $g(x)$ represents the original graph of $f(x)=|x|$ that has been moved 6 units to the right and 2 units up. The vertex, which was originally at $(0,0)$, is now translated to $(6,2)$. This vertex is the lowest point on the graph of $g(x)$, and understanding its new position is key to determining the new range. The transformation process is all about understanding how these simple algebraic changes to the function's formula directly correspond to predictable geometric movements on the graph. It’s like giving the graph new coordinates while keeping its fundamental shape intact.

Analyzing the New Function's Range: The Impact of Shifts

Alright guys, we've got our original function $f(x)=|x|$ and our newly transformed function $g(x) = |x-6| + 2$. Now, the million-dollar question: what happens to the range? Remember, the range is all about the possible output values, the y-values. For the original function $f(x)=|x|$, the lowest output was 0, and it went up to infinity. For our new function $g(x) = |x-6| + 2$, let's analyze its behavior. The term $|x-6|$ will always produce a non-negative output, just like $|x|$. Its minimum value is 0, which occurs when $x-6=0$, or $x=6$. So, the minimum value of the entire expression $|x-6|+2$ will happen when $|x-6|$ is at its minimum, which is 0. When $|x-6|=0$, the value of $g(x)$ is $0 + 2 = 2$. This means the lowest possible output for $g(x)$ is 2. Since $|x-6|$ can increase indefinitely as $x$ moves away from 6 (in either direction), the term $+2$ means that $g(x)$ can also increase indefinitely. Therefore, the range of the new function $g(x)$ is all real numbers greater than or equal to 2. We can express this as ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 2}$ or in interval notation as $[2, ext{∞})$. This is a direct consequence of the upward vertical translation of 2 units. The horizontal shift to the right by 6 units affected the domain where the minimum occurred (it shifted from x=0 to x=6), but it did not alter the minimum value of the function itself. The vertical shift, however, directly raises the entire graph, including its minimum point, thus changing the range. It's a clear demonstration of how vertical transformations directly impact the range, while horizontal transformations primarily affect the domain.

Comparing the Ranges: What Statement Holds True?

Now, let's put it all together and compare the ranges of the original function $f(x)=|x|$ and the transformed function $g(x) = |x-6| + 2$. We found that the range of $f(x)=|x|$ is ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 0}$. On the other hand, the range of $g(x) = |x-6| + 2$ is ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 2}$. Clearly, these two sets of possible output values are not the same. The original function can output any non-negative number, starting from 0, while the new function can only output numbers greater than or equal to 2. Therefore, any statement claiming the ranges are the same is incorrect. Let's look at the options provided (or implied by the question structure). Option A likely suggests the range is the same for both and is ${y ext{ | } y ext{ is a real number}}$, which is wrong on two counts – the ranges are different, and the original function's range isn't all real numbers. The correct comparison must acknowledge this difference. The lowest value the function can take has changed due to the upward shift. The horizontal shift moved the graph left or right, impacting where the minimum occurred, but the vertical shift directly determined the new minimum value. So, the true statement must reflect that the ranges are distinct, with the new function having a higher minimum output value. The range of $f(x)=|x|$ is $[0, ext{∞})$ and the range of $g(x)$ is $[2, ext{∞})$. These are fundamentally different sets of numbers. The difference in the ranges is precisely due to the vertical translation. This detailed breakdown should make it crystal clear why the ranges differ and how the transformations specifically cause this divergence. Keep practicing with different transformations, guys – it’s the best way to build your intuition!

Conclusion: Range Reflections

So, there you have it, mathletes! We took the absolute value function $f(x)=|x|$, translated it 6 units right and 2 units up to get $g(x) = |x-6| + 2$, and thoroughly analyzed their ranges. We confirmed that the original function $f(x)=|x|$ has a range of ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 0}$, while the transformed function $g(x)$ has a range of ${y ext{ | } y ext{ is a real number, } y ext{ ≥ } 2}$. This means the statement that is true about the range of both functions is that their ranges are different. The vertical translation upwards by 2 units directly shifted the entire range upwards by 2 units, changing the minimum possible output value from 0 to 2. The horizontal translation, while changing the x-coordinate of the vertex, did not affect the set of possible y-values the function could output. It’s a perfect illustration of how different types of transformations have distinct effects on a function's properties. Keep exploring, keep questioning, and keep transforming those functions! Until next time, stay curious!