Provide a short answer to each question. What is the domain of th...

Question

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

Accepted Answer

Hey, everyone here we are asked to identify the domain and range of the following function F of X is equal to 14 divided by X. Here we have four answer choice options, ABC and D, each of which state a differing domain and range in interval notation. So here we can begin this problem by first finding our domain. First, we need to recall that the domain is the set of all X values for which our given function is defined. So looking at our given function, we see that X is in the denominator. Therefore, we can note here that X cannot be equal to zero because that causes us to divide by zero. Thus results in our function to be undefined. However, if we were to input any other value of X into our function, we see that our function is in fact defined. So we can conclude that all values of X fit within our domain aside from X being equal to zero, since that causes our function to be undefined. So with this, we can write out our domain as going from negative infinity to zero and from zero to positive infinity. Now infinity is always excluded and therefore it gets closed by parentheses. And here we also determine that zero is excluded. Since when X is equal to zero, our function is undefined. So we can also close our zeros by parentheses. So here we have already identified our domain for our given function. And now we can move on to defining our range. Well, first, we need to recall that the range is the set of all Y values at which our function is defined. So we first want to consider as X approaches infinity and we see as X approaches infinity, we input larger and larger X values into our fun. And so the denominator grows larger, thus resulting in small smaller values from F of X. So we can conclude as X approaches infinity that F of X approaches zero. Now considering when X approaches negative infinity, we see the same result where the denominator grows larger and thus F of X also approaches zero. Now, with these two pieces of information, we see we have a line at Y is equal to zero. And this line serves as a horizontal Asymptote since our function will never cross this line. And therefore, we also know that zero cannot be included in our range because again, our function does not cross Y is equal to zero. So now we also want to consider as X approaches zero. Now there are two cases to consider with this. And the first we can consider is from the positive X axis or X approaches zero from the right side. And as X approaches zero, we see that F of X approaches infinity. And this is because we get smaller and smaller values of X substituted in, thus, resulting in larger and larger values from F of X. So F of X approaches infinity. Now, similarly from the left side or from the negative X axis, we see F of X approaches now negative infinity. So now with all of this information, we have found as X approaches positive nega and negative infinity, as well as zero from the right and left side, we can write our range as going from negative infinity to zero and from zero to positive infinity. Now, as we recall, infinity is always closed by parentheses. And here we also determined that zero is not included in our range since that is where we have a horizontal asym to. So our zeros also get closed by parentheses. So here we have now identified our domain and range for our given function. So now we can visualize our domain and range by creating a rough sketch, a rough sketch of our function. So now drawing our rough sketch, we start with a vertical line to represent our Y axis and then a horizontal line for our X axis. And now if we look at the first part of our domain and range, we see we will have negative X values and negative Y values respectively. So we are going to be in the third quadrant. And now looking at the domain, we go from zero to negative infinity and we also go from zero to negative infinity for the Y values or for the range. And now looking at the second part for the domain and range, we have positive X and Y values. So we will be in the first quadrant going from zero to positive infinity for the X values and from zero to positive infinity for the Y values. So now we have visualized and confirmed our domain and range with our rough sketch of the function. And so we are left here with answer choice. A where again, we have a domain going from negative infinity to zero and from zero to positive infinity and a range from negative infinity to zero and from zero to positive infinity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

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