State the asymptote of the function g(x)=6^x-9

Asymtote is at y=-9 . Remember for Exponential functions you can tell the aymptote by looking at the value which gives the Vertical translation up or down. In this case the vertical translation is 9 units down so therefore the asymptote is y=-9.

34

To determine the asymptote of the function g(x) = 6^x - 9, we need to observe the behavior of the function as x approaches positive or negative infinity.

First, let's consider x approaching positive infinity. As x becomes very large, the term 6^x will also grow exponentially. Since the exponent is positive, the base 6 raises to larger and larger powers, resulting in an infinitely large positive value. As a result, the function g(x) will approach positive infinity as x approaches positive infinity.

On the other hand, if we consider x approaching negative infinity, the exponent becomes increasingly negative. When a positive number is raised to a negative power, it yields a fraction or decimal approaching zero. So, as x tends to negative infinity, the term 6^x will approach zero. Subtracting 9 from this value, g(x) will approach -9. Hence, the function g(x) will approach -9 as x approaches negative infinity.

To summarize:
- As x approaches positive infinity, g(x) approaches positive infinity.
- As x approaches negative infinity, g(x) approaches -9.

Therefore, the horizontal asymptote of the function g(x) = 6^x - 9 is y = -9.