In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2...

Question

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+1) − 2

Accepted Answer

Hello everybody, everything. All right. Today, today we're going to be looking at this question that states graph the given rational function using the transformation of app of X is equal to one divided by X where the given function is G of X is equal to one divided by open parentheses X plus six and separate from the fraction minus one. Now, the gens provided are all sets of graphs showing both F of X and GEO X with certain points. So choice A F of X has a point at one comma one and G of X has a point at negative five comma two being as F of X with a point at one comma one and G of X with a point at seven comma zero C F of X has a point at one comma one and G FX has a point at seven comma two and D F of X has a point at one comma one and G of X has a point at negative five comma zero. Now, since they specifically asked to use the transformation of our F of X function, then we have to keep in mind different rules of transformations that we know. So I'm going to write the specific rules that we will be applying in this scenario. And I'll say that the graph of Y equals F of X plus C is the graph of Y equals F of X shifting C units left. So C here would be a constant or any number. And I'll also say that the graph of Y equals F of X and separate from the efax minus C is the graph of Y equals F for max shifting C units down once again, where C is a constant or a number. So I specifically mentioned these rules because that's what will apply to our functions. And I'll explain now. So let's say if F of X is equal to one divided by X and GEO is equal to one divided by open parentheses, X plus six, close parentheses and separate from the fraction minus one, then G of X is equal to F of open parentheses, X plus six closed parentheses minus one. So we compare this to what we already know with the rules that we have. And this means so therefore, F of X graph shifts six units left and one unit down. And we obtain all that information just from looking at our F of X function and our G of X function as well as keeping in mind the different rules of transformations that we know. So we are applying both the Y equals F of X plus C and the Y equals F of X and separate from that minus C rules. So with this in mind, we know that we will be shifting six units left and one unit down. So we see that for all the answer choices, alpha X has a point at one comma one. So we have a point at one comma one that if we shift six units left and one unit down, that would turn into one. So if we have, if we're moving left, we're changing the X value. So we'll have one. And since we're moving to the left, it's one minus six. And as for the Y value, we're moving one unit down. So we're also subtracting. So we'll have one minus six comma one minus one as our new point after doing the transformation, which will give the point negative five comma zero. So we have to see what graph shows F of X shifting six units left and one unit down or changing from the 10.1 comma one to a point and negative five comma zero. And if we look at our answer choices, we see that answer choice D does exactly that they give us the graph of F of X and the graph of GEO X and they give us points with F of X at one comma one and G of X at negative five comma zero. So answer choice D would be the answer for this scenario. So I hope that was helpful. And until next time.

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+1) − 2

Key Concepts

Transformations of Functions

Vertical and Horizontal Shifts

Graphing Rational Functions

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