C. Horizontal asymptote at .
We are asked to choose the option that best describes the asymptote of an exponential function of the form .
We know that exponential functions always have horizontal asymptotes. The horizontal asymptote for an exponential function in form is at .
Upon looking at our given function we can see that value of 'c' is 0, therefore, our given function will have a horizontal asymptote at and option C is the correct choice.
Horizontal asymptote at y=0
It depends on the value of b
Exponential functions are defined for positive values of the base [ŧex] b [/tex]. Also, they're not defined if , or at least let's say that this is a trivial case, since it is the constant function 1.
If the base sits between 0 and 1, the exponential function is constantly decreasing. The explanation is quite intuitive, assume for the sake of simpleness that b is rational. If it sits between 0 and 1, it means that it can be written as a fraction p/q, with p<q. So, if you give a large exponent to b, you obtain
On the other hand, if you consider negative exponents, you switch numerator and denominator and then raise to the same exponent the fraction q/p, which gets larger and larger.
So, if 0<b<1, we have
and thus 0 is a horizontal asymptote as x tends to (positive) infinity.
This case is very similar, except all roles are inverted. Now you start with a fraction p/q where p>q. So, with positive, large exponents you get
And as before, negative exponents switch numerator and denominator, so the fraction becomes q/p and thus you have
So, again, 0 is a horizontal asymptote, but this time for x tending towards negative infinite.
We are given exponential function : . Here x is the power (exponent) of b.
Please note: For any value of x for the exponential function, we can't get value 0.
Now, let us explain asymptote.
Asymptote is the line (vertical or horizontal) to which graph is approach and never meet.
For the exponential function for any value of x, y can't be equal to 0.
Therefore, graph would approach to y=0.And the horizontal asymptote of the exponential function f(x)=b^x is y=0.
The graph of an exponential function of the form is a graph which gets close to the x-axis but does not touch it.
Therefore, an exponential function of the form has a horizontal asymptote at y = 0.