Decreasing Exponential 4Dffdc

1. The problem is to understand the behavior and graph of the function $$f(x) = b^x$$ where $$0 < b < 1$$. 2. The formula for an exponential function is $$f(x) = b^x$$ where $$b$$ is the base and $$x$$ is the exponent. 3. Important rules for exponential functions: - If $$b > 1$$, the function is increasing. - If $$0 < b < 1$$, the function is decreasing. - The function is always positive, so $$f(x) > 0$$ for all real $$x$$. - The horizontal asymptote is $$y = 0$$. 4. Since $$0 < b < 1$$, the function decreases as $$x$$ increases. 5. As $$x \to \infty$$, $$b^x \to 0$$, so the graph approaches the x-axis but never touches it. 6. As $$x \to -\infty$$, $$b^x \to \infty$$, so the graph increases without bound on the left side. 7. The graph is always above the x-axis and is a smooth curve decreasing from left to right. Final answer: The graph of $$f(x) = b^x$$ for $$0 < b < 1$$ is a decreasing exponential curve above the x-axis, approaching $$y=0$$ as $$x$$ increases.