In Exercises 19–32, find the (a) domain and (b) range.
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Question

In Exercises 19–32, find the (a) domain and (b) range.

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𝔂 = -1 + ∛ 2 - x

Accepted Answer

Welcome back, everyone. Find the domain and range of the function G of X equals 2 minus cubic root of X minus 3. So let's begin with the domain of this function, and we have to recall that the domain of a function corresponds to all of the X values for which the function is defined. So we only have one term that has X, and that's cubic root of X minus 3, right? Let's ignore the negative sign for now and let's say that we are simply analyzing cubic root of x minus 3. Let's recall that whenever we have an odd root, well, essentially the term under the radical can be any term that we want, right? We can take a cubic root. Of a negative value and a positive value as well. So essentially we can say that the domain of cubic root of X is X belongs to all real numbers. So in this case, the only difference is that we have a horizontal shift. However, we can say that x minus 3 belongs to all real numbers. And therefore we are simply adding 3 to both sides, and this means that X belongs to all real numbers. There is no change due to the horizontal shift, right? If we, if X minus 3 belongs to all real values, then X also belongs to all real values, and definitely that's true because we can say cubic root of any number, any negative or positive number. So we have our domain. Now what about the range? Now for our range, we can either analytically think about the cubic root function. Or we can analyze several values. So basically, if X is equal to 3, well in this case, we have the horizontal shift, 3 units, the rights, we can actually modify our curve. And now, if X is equal to 3, then our cubic root term is equal to 0, but when we keep increasing X, we're taking cubic root of a larger and larger number. So basically, the larger the number gets, the closer we get to an infinitely large number. On the other hand, if we take X, which is less than 3, and keep decreasing the value of X, we're approaching negative infinity, right? So once again it depends on the magnitude of the actual number, but basically our cubic root function extends from negative infinity up to infinity. Now the negative sign is basically a reflection. Right? Relative to the x axis and since we're adding to, it's just a vertical shift up, so it doesn't really change anything regarding the range itself. We can basically say that the Y value also belongs to all real numbers. So now let's conclude our findings. Let's begin with the domain or the domain. X belongs to all real numbers in an interval notation from negative infinity up to infinity and now for the range. The y value belongs to all real numbers or from negative infinity up to positive infinity, and that would be our final answers. So let's label them. And thank you for watching.

In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x

Key Concepts

Domain

Range

Cube Root Function

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