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

Question

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

Accepted Answer

Hello everybody. Everything right. Today, today we're going to be talking about this question that states provide the domain and range of the function by graphing it where our function is FX is equal to open parentheses. One divided by 11, close parentheses to the exponent X plus seven. Now for a question like this where I see my app of X function. The first thing I want to ask myself is if I can get this to its simplest form or if it's already at its simplest form. In this case, it's not, I can determine what I'm going to call my parent function, which is basically the most basic way of writing a function like this before adding things like the plus seven to the exponent. In this case, I'll say that my parent function will be GEO XX is equal to open parentheses, one divided by 11 closed parentheses to the exponent X. So this is my parent function and will be, it will serve as my parent graph. So I'll say that the G of X graph looks like open parentheses. One divided by a where A could be in number closer is used to the exponent X. So what will that parent graph look like? Well, I'll draw a general way of it. So I'll draw a Y axis and an X axis. And I know that the graph of one divided by A to the exponent X has a Y intercept at one and has the X axis as a horizontal ascent tode. So the graph of a function in that general form will look like a stretched out L. So now that I have that established, I can see how I move from my parent graph to my current function which is FX. So F of X is equal to open parentheses, one divided by 11 closed parentheses to the exponent X plus seven. So all you do to get from the parent function to the F F to the parent from the parent function G of X to our current function F of X. All you have to do is add a plus seven to the R exponent, which means that G of X shifts seven units left. So that is exactly, that's what the plus seven will do to our graph. It's going to shift it seven units to the left. Now let's try some points. So we can say X is equal to negative seven. And if you plug in X is equal to negative seven, we have an exponent of zero and anything to the exponent zero is one. So if X is equal to negative seven R Y will be equal to one or the point negative seven comma one. And we can also plug in X is equal to negative eight. And if we do that, we'll have an exponent of negative one, which means that we'll take the reciprocal of our fraction, which means that Y is equal to 11 or the point negative eight comma 11. So now that I have that information, I can draw a general form of my graph. So uh once again, draw a Y axis and an X axis and I can begin plugging in points that I know. So I know that I'll have a point at an X of negative seven and an X of negative eight. And once again, I had a negative seven comma one. So I'll draw that point and negative eight comma 11. So I'll draw that point and I'll connect both of those with a curve. And once again, I know that I'll be approaching the X axis as a horizontal Asymptote and never touch it. And the horizontal Asymptote is never affected because all we did to the graph of our parent function was shifted seven units to the left, which is why our initial Y intercept, which was zero comma one has now that point has now shifted to negative seven comma one. So once again, we'll have kind of a curved L it's the same shape as our parent function. Just moved seven units to the left. So now we can talk about the domain and range, the domain having to do with the possible X values of our function. So we just look how it moves from left to right. And we can see that it moves infinite to the left and infinite to the right. So our domain is from negative infinity to positive infinity. And now we're looking at the range, we look at possible Y values. So how our function moves up and down the Y axis. So if we start at the bottom of a graph, we see that we approach zero but never touch it. So that's our starting point and it'll be an open parentheses because we never touch it. So it'll be open parentheses, zero, comma so from 02 and as we move up, we go towards positive infinity. So our range will be zero comma positive infinity. And just like that, we have obtained a graph of our function F of X as well as identified 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)

Key Concepts

Graphing Exponential Functions

Domain of a Function

Range of a Function

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