Graph each function. Give the domain and range. See Example 3....

Question

Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)^-x

Accepted Answer

Hello everybody. I today, today we're going to be looking at this question that states provide the domain and range of the function by graphing it where our function is F of X is equal to open parentheses. One divided by 11 closed parentheses to the exponent negative X. Now, when it comes to a question like this, I want to try to identify if there is a parent function first and see what changes are applied to that parent function. So in this case, that parent function, I will label it as G of X is equal to open parentheses. One divided by 11 to the positive X exponent. Now all we do to go from that parent function to our F of X function is add the negative in front of the exponent. So once again, our F of X is equal to open parentheses, one divided by 11 closed parentheses to the exponent negative X. So what that means is that we are the graph of G of X reflects about the Y axis. So now let's try to determine different qualities and rules that we know of our GEO X our parent function graph. So G O X has a horizontal, has a horizontal as to at the X axis. So the X axis functions as a horizontal asym tilt for this graph. And we should also note that any graph in the form in the form of open parentheses, one divided by a where a is a constant closed parentheses to the exponent X. Well, we have a general shape and I'll draw that general shape. So we'll have a Y axis and an X axis. Once again, we mentioned that the X axis will function as a horizontal aceto. And then we'll have a general form where we start from behind the Y axis go down and we're slowly turning up to a certain point and then we turn and intersect the Y axis and then approach the X axis but never touch it. So it's kind of like an L, a bit of a stretched out L. So the lines aren't specifically straight. It's more of a curve in the shape of an L. So now that we have that general information, let's plug in certain points for our F of X value. So now we're going to focus on our F of X function. So we graph F of X is equal to over parentheses, one divided by 11 close parentheses to exponent negative X. So now that we're looking at, we're done looking at rules with our G of X function or parent function, let's focus on our actual function. So we'll try different points. So let's say F of one is equal to one divided by 11 to the exponent negative one, which is equal to 11 or the 0.1 comma 11. And then we can also try a point at zero or X of zero. So we'll have F of zero is equal to open parentheses. One divided by 11 to the exponent zero, which will be equal to one or the 0. comma one which will function as R Y intersect. So now that we have all these rules, we can go ahead and draw a general shape of our graph. So it will draw a Y axis and then X axis. And now we know that we intersect this Y axis at a Y of one, we just obtained that point. And we also know that at an X of one, we reach a Y value of 11. So we'll plug that point in as well. And we know that the horizontal ascent tode is still a, the X axis because we didn't change our, our graph, our parent function didn't move up or down, but it is reflected about the Y axis. So where we had in our parent function, we extended upwards in quadrant two, we will now be extending upwards in quadrant one since we're reflecting on upon the y axis. And now we will extend infinitely to the left and approach the X axis on the left instead of on the right. So now it'll be like a backwards L in a backwards L with, in the shape of a curve. So once again, the lines aren't straight, but it has the general shape of an L but a backwards L where they why intercept at zero comma one and another point at one comma 11. So now we can talk about the domain and range. So the domain has to do with the possible X values of our graph. And in this case, we see that our graph moves infinitely to the left and infinite to the right. Therefore, our possible X values will be will go from negative infinity to positive infinity in our range has to do with the possible Y value. So we have to see how a graph moves along the Y axis. So up and down and we see that where it starts, we are approaching the X axis but never touching it. And we don't go under the X axis so that X axis value in Y will be zero. So that is uh the X axis has a Y value of zero. So our range will be from 02. Then we see that our graph extends infinitely upwards. So it'll be from zero to positive infinity. And the range will have an open parentheses and a closed parentheses because we never touch zero and we never touch infinity. If we were to touch zero, then that open bra uh parentheses will be an open bracket. So if you touch one of the ranges, one of the ends of the range, it will have a bracket. But since we don't do that, we will have a parentheses. And just like that, we have been able to graph our function F of X as well as determine its domain and range. So I hope that was helpful. And until next time.

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