Graph the given function. g ( x ) = 4 − x − 1 g\left(x\right)=4^{-x}-1 g ( x ) = 4 − x − 1

In this problem, we're asked to graph the given function. And here our given function is g of x is equal to 4 to the power of negative x minus 1. Now let's go ahead and just jump right into graphing here starting with step number 0 which is is to identify and graph our parent function. Now since here we have 4 to the power of negative x minus one, our parent function is going to be f of x is equal to 4 to the power of x because that's the function that I'm seeing here. Now we can go ahead and plot this using the points negative one, one over b which would be 1 over 4 and then 1 and then finally one 4. So plotting those points, negative 1, 1 fourth, really small close to the x axis there, 1 and 14. Now, this graph is rather steep but we can still connect it and then, of course, getting really close to our x axis there because we have a horizontal asymptote that we can go ahead and plot at y equals 0. So plotting that using a dash line, now we have completed step 0. We can go ahead and move on to step 1 and start transforming this parent function. So we want to go ahead and shift our horizontal asymptote to y equals k, which here I have this negative one which represents my k. So I'm going to shift my horizontal asymptote to y equals negative one. And I can go ahead and plot that using a dash line because this is an asymptote. Now moving on to step number 2, we need to decide if there is a reflection happening. Now remember reflection happens if I have a negative inside or outside of my function which up here I have 4 to the power of negative x so I do indeed have a reflection here and this reflection since it is inside of my function with that x is going to be over the y axis. So let's go ahead and take our test points from our parent function and reflect those over our y axis. So starting with this point up here, I'm going to go ahead and reflect that over ending up right here. And then my point 1, I actually can't reflect that over the y axis, is gonna stay in the same spot. And then finally, this point negative 11 4th, I can go ahead and reflect that point as well. So I have my new points, but I'm not done with this step 2. I need to also shift these points by h k. Now here, h is just a 0. I don't have to worry about a horizontal shift, but I know my value for k is negative 1. So I do need to shift these points down by 1 point. So let's go ahead and do that. So shifting this point down, I'm gonna end up right here, shifting this point down as well, and then finally up here. So I'm gonna go ahead and just have those points left, my new points. And now I am done with step 2 and I'm actually done graphing. I can go ahead and connect everything here. So I'm of course going to be approaching this asymptote and then of course connecting on the other side. So my new function is this function in blue. We see that it is reflected over the y axis. It has been flipped that direction and and it has been shifted down. So now we want to identify our domain and our range here. Our domain is always the same. It's always going to be all real numbers so nothing to worry about there And then I need to look for my range if my graph is above or below my asymptote. Now my asymptote is here at y equals negative one, and my graph is above that. So that tells me that my range is going to go from k, my value for k, my asymptote, which here is at negative one, to infinity. So my domain is all real numbers and my range goes from negative one to infinity. Now that we've graphed that, thanks for watching and let me know if you have any questions.

6. Exponential & Logarithmic Functions

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6. Exponential & Logarithmic Functions

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Multiple Choice

Graph the given function.
$g\left(x\right)=4^{-x}-1$

Verified step by step guidance

Identify the function type: The given function is an exponential function of the form g(x) = 4^{-x} - 1.

Determine the horizontal asymptote: For the function g(x) = 4^{-x} - 1, the horizontal asymptote is y = -1.

Find the y-intercept: Set x = 0 in the function to find the y-intercept. g(0) = 4^{0} - 1 = 1 - 1 = 0, so the y-intercept is (0, 0).

Analyze the behavior of the function: As x approaches positive infinity, 4^{-x} approaches 0, so g(x) approaches -1. As x approaches negative infinity, 4^{-x} becomes very large, so g(x) becomes very large.

Plot key points and asymptote: Plot the y-intercept (0, 0) and the horizontal asymptote y = -1. Sketch the curve starting from the y-intercept, approaching the asymptote as x increases, and rising steeply as x decreases.