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,

0 < b < 1

and

b > 1 .

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 $b^{- x} = \frac{1}{b ^{x}} .$ 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 $0 < b < 1$

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 $b^{- x} = \frac{1}{b ^{x}} .$ 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.

Case b>1

Case 0<b<1

Conclusion

Case $b > 1$

Case $0 < b < 1$

Exercise

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