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 - 2

Accepted Answer

Hello everybody. I have everything right today. Today we're going to look 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 X and separated from all that minus eight. Now, when looking at a function like this or a question like this, I want to ask myself if the F of X function that they gave me is in its most basic form. And what I mean by that is, is there a parent function here? And in this case there is, and I'll label that parent function G of X. So I have to think what would that parent function be? Well, the most basic form here, I see the minus eight separated from the rest of my function. That is a transformation. So that's something that is applied to the parent function that will move it. So if we take that minus eight out, we'll be left with our parent function with this, which is G of X is equal to open parentheses. One divided by 11 closed parentheses to the exponent X. Now we have our parent function is a fraction to an exponent. And so let's recall that the graph of a function in the general form open parentheses, one divided by a, where A is a constant closed parentheses to the exponent X that graph looks like. And I'll draw a very general way of that graph. So I'll have a Y axis and an X axis. And now when we have a graph of a fraction to an exponent X, we know that it'll have a Y intercept at zero comma one and that the X axis will function as a horizontal Asymptote. So from that Y axis, I'll extend downwards until I'm close to the X axis. And at the same time, I'll be moving to the right and before the Y, the Y intercept um moving upwards to the left as a curve. So the graph of, of our fraction one divided by a, to the exponent X looks kind of like a curved L. So with that in mind, now let's think about how we get from the parent function to our current function. So our E X function is equal to open parentheses, one divided by 11 closed parentheses to the exponent X. And now we add the minus eight separate from all that. And this means that, that minus eight means that G of X shifts eight units down. So that is what happens when we have the minus eight and I'll make a note here. I'll write H A which means horizontal asymptotes at Y equals zero just making sure we have that note of our parent function. So now we go back to our F function and now that I see how we get from the parent function to our current F of X function just by shifting it eight units down. Now I'm gonna try different points. So first I'll try F of zero, which means that an X of zero. So F of zero is equal to open parentheses, one divided by 11 closed parentheses to the exponent zero. Now, because I'm plugging in the zero for the X and separate from all that a minus eight. Now that would be equal to one minus eight because anything to an exponent zero is one and one minus eight equals negative seven. So we'll have the 70.0 comma negative seven. We add an X of zero, we have a Y of negative seven and then I'll try a different point this case F of negative one. And I'll plug that into our F of X equation. So I'll have open parentheses, one divided by 11 closed parentheses to the exponent negative one and separate from that A minus eight. Now that'll be equal to 11 minus eight. Why? Because we have a negative one in our exponent. And then if we have a negative one in the exponent, we have to take a fraction of whatever is being raised to that exponent. In this case, we have a fraction being raised to a negative one. So you just take the reciprocal, if you have one divided by, by 11, you now have 11. So 11 minus eight is equal to three, which gives us the point negative one comma three where at an X of negative one, we have a Y of positive three. So now let's continue with our eve function. So our horizontal as so will also shift eight units down and is now at Y equals, and let's think about it. We had a, a horizontal Asymptote out of Y equals zero. And we shipped that eight units down. Now we have one at Y equals negative eight. So that's the reason as to why we look at the parent function first and see how we change that. Now we know that we'll have a horizontal as at A Y of negative eight and we know of two other points that we have in our graph. So we can go ahead and graph a general idea of it. So once again, we'll draw a Y axis and an X axis and we'll plug in the different points that we know. So we know we're gonna have a point at a Y of three. So we draw dash on our Y axis at a uh what we are going to call three. And we also know we're gonna have a point at an X of negative one. So we do do the same on the X axis and we put a point at negative one comma three. Next, we know that our Y intercept was going to be at zero comma negative seven. So we draw a dash at negative seven on the Y axis and we put a point there at zero comma negative seven. And now we know that our horizontal Asymptote will be at negative eight. So we can label negative eight and draw kind of a dashed line just to show that there will be a horizontal ascent to there. So now we uh keeping in mind that this was a curve looking at our paragraph. So we can just connect the dots, connect the points that we have and curve at around the Y intercept as we approach the horizontal ascent and negative eight extend infinitely to the right, never touching that negative eight value. So once again, our graph will look like a curved L and now we can talk about domain and range. So when talking about domain arrange the domain has to do has to do with the possible X values of our graph. So basically, you just want to look at how your graph behaves left and right, since moving left and right is how you look at the X axis. So let's look at our graph and how it moves to the left, the left. First, we see our, our graph curves upwards. And although to the left, it moves at a much slower rate than to the right. It is still infinitely going to the left. Although it is very slow, it will always continue to move to the left. There is nothing stopping it. So that is where Nega, that's our, our starting point, negative infinity is our starting point. So we draw, we write uh open parentheses, negative infinity comma and you can think of that comma as a two. So we are not the number but the word T O. So you can think of it as writing negative infinity two and your ending point in the domain. So we have negative infinity comma. And then we look at how our graph behaves on the X axis to the right. And we see that we curve, the curve extends to the right at a much faster rate than to the left. But it's still the same. It, it extends infinitely to the right. Nothing stopping it. So we'll have negative infinity comma positive infinity or from negative infinity to positive infinity. And then we write a closed parentheses. Now, for the range, we have to look at how our graph moves up and down. So how is, how does it behave along the y axis? So what would be our starting point here? Well, in this case, we start from the bottom up. So we know we have a horizontal Asymptote at negative eight. Now that means that our graph will approach negative, negative eight, but never touch it and never pass it. So because we never touch it, we write an open parentheses. If we did touch it, this would be a bracket. But it, since we don't, we write an open parentheses and it's the same concept. So from a, a starting point comma or to the endpoint. So in this case, our starting point would be negative eight and like we said, open parentheses, negative a comma. And now we see how a graph moves upwards and we know that it extends infinitely upwards as there's nothing stopping it, stopping it, no point or no, no other as to stopping it. So it'll be from negative eight to positive infinity. And then we close the parentheses again. And once again, we use the parentheses for the closing because infinity is a value that you will never touch. So just like that, we've drawn a graph of our function F M X as well as it's figured out its domain and its range. So I hope that was helpful. And until next time.

Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)^x - 2

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Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)x - 2

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Graph each function. Give the domain and range. See Example 3. ƒ(x) = (1/3)^x - 2