The function $f(x) = 5^x - 4$ is an exponential function with a base of $5$. Here's some important information about it:

1. Domain: The domain of $f(x)$ is all real numbers, $(-\infty, \infty)$, since exponential functions are defined for all real $x$.

2. Range: Since $5^x > 0$ for all $x$, and $-4$ shifts the function down by 4, the range of $f(x)$ is $(-4, \infty)$.

3. Intercepts:

   • y-intercept: When $x = 0$, $f(0) = 5^0 - 4 = 1 - 4 = -3$. So, the y-intercept is at $(0, -3)$.
   • x-intercept: Set $f(x) = 0$: $5^x - 4 = 0$, or $5^x = 4$. Solving for $x$, $x = \log_5(4) = \frac{\ln 4}{\ln 5}$.

4. Behavior:

   • As $x \to \infty$, $f(x) \to \infty$.
   • As $x \to -\infty$, $f(x) \to -4$. This indicates a horizontal asymptote at $y = -4$.

5. Graph: The graph of $f(x)$ is an exponential growth curve shifted downward by 4 units, starting near $y = -4$ for large negative $x$ and growing without bound for large positive $x$.

Would you like me to plot this function or explore its properties further?

Related Questions:

1. What is the derivative of $f(x)$, and what does it tell us about the slope of the curve?
2. How does changing the base of the exponential function affect the graph?
3. What is the inverse of $f(x) = 5^x - 4$?
4. How would a vertical or horizontal transformation of this function affect its intercepts?
5. Can we solve equations involving $f(x)$ for specific values of $x$?

Tip:

Exponential functions grow (or decay) faster than polynomial functions for large $|x|$. Understanding the base and transformations helps interpret the graph efficiently.