We can rewrite a negative exponent power as fraction with a a positive exponent power on its denominator b^(- x) = 1b^x.
No, see solution.
Practice makes perfect
Notice that for a graph of f(x)=b^x to have an x-intercept, the value of the function has to be zero at some point. Recall that we are considering positive values of b.
b^x=0
There are two cases to consider, 01. We will consider both cases, one at a time.
Case b>1
Let's consider the possible values of x, and how they affect the function.
If x=0, then b^x= 1.
If x>0 the function will be increasing, since any number greater than 1 becomes greater when multiplied times itself.
For negative values of x we can rewrite the power using the b^(- x) = 1b^x. This way, the denominator will have a positive exponent. In consequence, the function value will be decreasing. Nevertheless, it will never be zero, as there is no number that can divide 1 to get 0 as result.
We can see this behavior depicted in the example graph below.
Case 0
Let's consider the possible values of x, and how they affect the function in this case.
If x=0, then b^x= 1.
If x>0 the function will be decreasing, since any number b such that 0 < b < 1 becomes smaller when multiplied times itself. However, it will never be zero, because a product is zero if and only if any of its factors are zero as well.
For negative values of x, we can rewrite the power using b^(- x) = 1b^x. This way, the denominator will have a positive exponent. Then, the denominator will decrease, and as a consequence the function value will increase.
We can see this behavior depicted in the example graph below.
Conclusion
An exponential function never touches the x-intercept no matter which value we use for the base, and no matter which value x takes. It will get closer and closer to 0 in one direction, but it will never reach it. This is know as asymptotic behavior.
