Compute $6/4$, $9/6$, and $13.5/9$. What do you notice about these ratios, and what type of function has a constant ratio between consecutive outputs?

If $g(x)$ is exponential with base $1.5$, write it as $g(x)=a(1.5)^x$. Use the value at $x=0$ from the table to find $a$, then look for that form among the choices, paying attention to the sign of the exponent.

Create a table in Desmos with $x$ values $0,1,2,3$ and corresponding $y$ values $4,6,9,13.5$ to represent the given data points.

Type each option as a separate equation, for example y = 6(1.5^x), y = 4(1.5^x), etc. Desmos will plot all four graphs on the same axes.

Look at which graph passes through all four data points from the table (the points in your table will appear on the graph). The function whose graph hits every point in the table is the equation that could define $g$.

Look at how $g(x)$ changes when $x$ increases by 1:

• From $x=0$ to $x=1$: $4$ to $6$ (change of $+2$)
• From $x=1$ to $x=2$: $6$ to $9$ (change of $+3$)
• From $x=2$ to $x=3$: $9$ to $13.5$ (change of $+4.5$)

The differences are not constant, so $g$ is not linear. For exponential functions, we expect a constant ratio between consecutive values, not a constant difference.

Now check the ratios between consecutive $g(x)$ values:

• $6/4 = 1.5$
• $9/6 = 1.5$
• $13.5/9 = 1.5$

The ratio is constant and equal to $1.5$, so $g$ appears to be an exponential function of the form

for some constant $a$.

For an exponential function $g(x) = a \cdot (1.5)^x$, plug in $x=0$ and use the table value $g(0)$:

• From the table, $g(0) = 4$.
• In the formula, $g(0) = a \cdot (1.5)^0 = a \cdot 1 = a$.

So $a = 4$. Keep this value in mind when comparing with the answer choices.

Compare $g(x) = 4(1.5^x)$ to the answer choices:

• It has coefficient $4$ (matching $g(0)=4$).
• It has base $1.5$ (matching the constant ratio of $1.5$ between table values).
• For a quick check, $g(3) = 4(1.5^3) = 4 \cdot 3.375 = 13.5$, which matches the table.

The equation that fits all these conditions is $g(x)=4(1.5^{x})$.