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.

Explore the exponential function f(x) = 3^x - 7 with detailed insights into its graph, domain, range, and vertical shift. Learn about exponential growth and how the function behaves as x varies. Suitable for high school students and those interested in advanced algebraic functions.

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