The function $f(x) = 6^x$ is an exponential function where the base is 6 and the exponent is $x$.

Key Points:

1. Exponential Growth: The function $6^x$ grows rapidly as $x$ increases because the base, 6, is greater than 1.

2. Domain: The function is defined for all real numbers $x$, so the domain is $(-\infty, \infty)$.

3. Range: Since $6^x$ is always positive for any real $x$, the range is $(0, \infty)$.

4. Y-intercept: To find the y-intercept, evaluate $f(x)$ at $x = 0$: $f(0) = 6^0 = 1$ So, the y-intercept is at $(0, 1)$.

5. Asymptote: The horizontal asymptote of this function is the x-axis, or $y = 0$, because as $x$ approaches negative infinity, $6^x$ approaches 0 but never actually reaches it.

Graphing the Function:

• The graph of $f(x) = 6^x$ is an upward-sloping curve that starts very close to 0 (on the y-axis) when $x$ is negative and increases rapidly as $x$ becomes positive.

Would you like to explore this function further, such as by finding the inverse or analyzing its graph more closely?

Related Questions:

1. How does the graph of $f(x) = 6^x$ compare to $f(x) = 2^x$?
2. What is the derivative of $f(x) = 6^x$?
3. How do you find the inverse of the function $f(x) = 6^x$?
4. Can you solve for $x$ when $f(x) = 36$?
5. How does changing the base (e.g., from 6 to 10) affect the shape of the graph?

Tip: In exponential functions, the base determines how quickly the function grows. A larger base results in faster growth.