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

Question

Provide a short answer to each question. What is the domain of the function $ƒ (x) = \frac{1}{x^{2}}$ ? 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 divided by X squared. Here we have four answer choice options, answer choice. A the domain goes from negative infinity to zero and from zero to positive infinity. And the range is the same as the domain answer B the domain is from negative infinity to positive infinity. And the range is from zero to positive infinity. Answer C the domain is from negative infinity to zero and from zero to positive infinity where the range is from zero to positive infinity. Answer D the domain is from negative infinity to zero and from zero to positive infinity. And the range is from negative infinity to zero. So before we can identify the domain and range, we first need to recall that the domain is the set of all X values for which our function is defined. And the range is the set of all Y values for which our function is defined. So beginning with binding our domain, we just need to notice that we have X squared in the denominator of our function. So we know that X cannot be equal to zero. Since if X is equal to zero, we will be dividing by zero. Thus resulting in our function being undefined. So we know that we can input any value into this function aside from zero. So this tells us that our domain is from negative infinity to zero and from zero to positive infinity. Now infinity is always excluded and therefore closed by parentheses. And in this case, zero is also excluded since it causes our function to be undefined. So it also gets closed by parentheses. So now we have already determined our domain of the given function, so we can move on to finding our range. So first, we want to assess our function as X approaches negative infinity. So as we input smaller and smaller values into our function here or larger and larger negative values, it results in a larger and larger denominator which causes F of X overall to become smaller and smaller. So that means as X approaches negative infinity, the F of X approaches zero. And now assessing the function's reaction as X approaches positive infinity, we see the same results, the denominator gets larger and larger, thus resulting in F of X getting smaller and smaller. So F of X again will approach zero. So since in both cases, F of XX approaches zero, this tells us that we have a horizontal Asymptote at X is equal to zero. And since we have an Asymptote at this point, this tells us that zero will not be in the range. And we also want to assess as X approaches zero, we can notice that F of X will approach positive infinity. So this tells us that the range is then zero to positive infinity where zero is not included and therefore closed by a parentheses. So to help understand our domain and range of our function or how our function really looks like we can create a rough sketch where this vertical line is our Y axis and then the horizontal line is our X axis. And now starting in the second quadrant, we can look at our first part of the domain where we go from negative infinity and we approach zero. And now looking at our range, we go from zero to positive infinity along the Y axis. And now repeating this process with the second part of our domain, we go from zero to positive infinity in the first quadrant and then go from zero to positive infinity along the Y axis. So again, we have this horizontal or horizontal Asymptote along the X axis and we also have a vertical Asymptote along the Y axis. And so with this visualization and our identified domain and range, we are left with answer choice C where again our domain ranges from negative infinity to zero and from zero to positive infinity. And our range is from zero to positive infinity. Thanks for watching I hope you found this video helpful and I'll see you again next time.

Provide a short answer to each question. What is the domain of the function $ƒ (x) = \frac{1}{x^{2}}$ ? What is its range?

Key Concepts

Domain of a Function

Range of a Function

Behavior of Rational Functions with Squared Denominators

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Provide a short answer to each question. What is the domain of the function ƒ(x)=1x2ƒ(x)=\(\frac{1}{x^2}\)? What is its range?

Key Concepts

Domain of a Function

Range of a Function

Behavior of Rational Functions with Squared Denominators

Watch next

Provide a short answer to each question. What is the domain of the function $ƒ (x) = \frac{1}{x^{2}}$ ? What is its range?