Answer:
![f(x) = 3 \cdot 0.2^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%203%20%5Ccdot%200.2%5Ex)
Step-by-step explanation:
![f(x) = 3 \cdot 0.2^x](https://tex.z-dn.net/?f=f%28x%29%20%3D%203%20%5Ccdot%200.2%5Ex)
Domain of f: "What values of x can we plug into this equation?" This makes sense for all real numbers so the domain is ![\mathbb{R}](https://tex.z-dn.net/?f=%5Cmathbb%7BR%7D)
Range of f: "What values of f(x) can we get out of the function?" From the graph we see we can get any real number greater than 0 out of the function by choosing a suitable x-value in the domain. The range is therefore
.
Continuity: Since the graph is one, unbroken curve (i.e. a curve that can be drawn in one movement without taking your pen off the paper). We see that "roughly speaking" the function is continuous.
Increasing or decreasing behaviour: For all x in the domain, as x increases, f(x) decreases. This means the function exhibits decreasing behaviour.
Symmetry: It is clear to see the graph of f(x) has no symmetry.
Boundedness: Looking at the graph we see it is unbounded above as when we choose negative values, the graph of f(x) explodes upwards exponentially. Choose a value of x, plug it in, next choose (x-1), plug this in and we observe
for all x in the domain.
The function is however bounded below by 0: no value of x in the domain exists which satisfies
.
Extrema: As far as I can tell, there are no turning points on the curve. (Is this what you mean by extrema?)
Asymptotes: Contrary to the curve's appearance, there are no vertical asymtotes for this curve. The negative-x portion of the curve is just growing so quickly it appears to look like an asymptote. There is a value of f(x) for all x<0. There is however a horizontal asymtote:
.
End behaviour: As
. As ![x \rightarrow -\infty, f(x) \rightarrow +\infty](https://tex.z-dn.net/?f=x%20%5Crightarrow%20-%5Cinfty%2C%20f%28x%29%20%5Crightarrow%20%2B%5Cinfty)