Use analytical and/or graphical methods to determine the large...

Question

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} $f\left(x\right)=\frac{1}{x^2}$

Accepted Answer

Welcome back, everyone identify the largest set of points where the function J of X equals four divided by X squared can have an inverse using analytical or graphical methods. And we are given for answer choices A from negative infinity to one and from one to infinity, B from negative infinity to zero, C from negative infinity to infinity, and D from negative infinity to zero and from zero to infinity. So first of all, let's define our function as we can see J of X is equal to four divided by X squared. Our first goal is to simply identify the domain of the function. We want to identify the values of X for which this function is defined. And because we have a rational expression, we have to equal that division by zero is undefined. So essentially X cannot be equal to zero. Any other value of X works for us, right? We can evaluate the value of J of X. So we have our domain and now we want to consider this function well, essentially if we split our domain from negative infinity 20, not inclusive, then this function would be increasing. Notice that X approaches zero right, from negative infinity to zero. So the value of X decreases and dividing four by a smaller number essentially results in a larger number. And the denominator will always be positive since we have X squared right. And now the next part from zero to infinity, this is where the function is going to decrease. We are increasing the value of X and the larger it gets the smaller the ratio becomes. So the function will be decreasing, right? We want to understand that an inverse is defined where the function is 1 to 1, meaning it must be monotonically increasing or decreasing within the interval. So the two intervals are the intervals where the function is 1 to 1, meaning we are going to use both of them. Therefore, the function will have an inverse in these two intervals, right? From negative infinity to zero and from zero to infinity. So essentially looking at the answer choices, we're going to conclude that the correct answer to this problem would be option D from negative infinity to zero and from zero to infinity. Non inclusive. Thank you for watching.

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Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.
{Use of Tech} $f\left(x\right)=\frac{1}{x^2}$