The question asks to identify the graph of $f(x) = \left(\frac{9}{5}\right)^x$.

Step-by-step Analysis:

1. Base Analysis:

   • The function $f(x) = \left(\frac{9}{5}\right)^x$ is an exponential function.
   • Since $\frac{9}{5} > 1$, this is an exponential growth function, meaning it increases as $x \to \infty$.

2. Behavior of Exponential Growth:

   • For $x = 0$: $f(0) = \left(\frac{9}{5}\right)^0 = 1$. The graph should pass through the point $(0, 1)$.
   • For $x > 0$: $f(x)$ increases rapidly because $\frac{9}{5} > 1$.
   • For $x < 0$: $f(x) = \left(\frac{9}{5}\right)^x = \left(\frac{5}{9}\right)^{-x}$, which decreases towards 0 as $x \to -\infty$.

3. Graph Features:

   • The graph will:
     • Pass through $(0, 1)$,
     • Rise sharply for $x > 0$,
     • Approach $y = 0$ (the x-axis) as $x \to -\infty$.

4. Choice of Correct Graph:

   • Graph [A] exhibits the described features: it grows exponentially for $x > 0$, passes through $(0, 1)$, and approaches $y = 0$ as $x \to -\infty$.
   • Other graphs ([B], [C], [D]) either depict decay, incorrect growth behavior, or do not pass through $(0, 1)$.

Conclusion:

The correct answer is [A].

Would you like a detailed explanation of exponential functions or a breakdown of the rejected graphs?

Here are 5 related questions to expand your understanding:

1. What is the general form of an exponential function?
2. How does the base of an exponential function affect its graph?
3. What is the significance of $f(0)$ in exponential functions?
4. How do exponential decay and growth differ graphically?
5. Can the range of an exponential function ever include negative values?

Tip: Always check the base of the exponential function to predict its growth or decay behavior!