The function you've provided is $f(x) = 3^x - 7$.

This function represents an exponential function where the base is 3 and the exponent is $x$, followed by a subtraction of 7 from the result.

Key Points:

1. Exponential Growth: The term $3^x$ indicates exponential growth as $x$ increases.
2. Vertical Shift: The subtraction of 7 indicates a vertical shift of the graph downward by 7 units.
3. Domain: The function is defined for all real numbers $x$, so the domain is $(-\infty, \infty)$.
4. Range: Since $3^x$ is always positive and gets larger as $x$ increases, the smallest value of $f(x)$ occurs when $3^x = 1$ (which happens when $x = 0$). Therefore, the range is $(-7, \infty)$.
5. Y-intercept: To find the y-intercept, evaluate $f(x)$ at $x = 0$: $f(0) = 3^0 - 7 = 1 - 7 = -6$ So, the y-intercept is at $(0, -6)$.

Would you like to go over the graph of this function, or do you have any other specific questions?

Related Questions:

1. How does the graph of $f(x) = 3^x - 7$ compare to $f(x) = 3^x$?
2. What is the behavior of $f(x)$ as $x$ approaches infinity and negative infinity?
3. How do you find the inverse of the function $f(x) = 3^x - 7$?
4. Can you solve for $x$ when $f(x) = 0$?
5. How would the function change if the base were different, say 2 instead of 3?

Tip: Understanding the transformation of the base exponential function $f(x) = b^x$ is crucial for graphing and interpreting more complex exponential functions.