All the $g(x)$ values in the table are positive. Which answer choices would always give negative values for $g(x)$?

After removing choices that give negative outputs, compare the two remaining bases. Which kind of base (greater than 1, or between 0 and 1) would make the function values get smaller as $x$ increases?

For the remaining choices, plug in $x=1$ or $x=-1$ and see which equation gives the same values as in the table.

In Desmos, type each option on its own line as a function of $x$:

• y = -25^x
• y = -(1/25)^x
• y = 25^x
• y = (1/25)^x

Click the plus (+) button and choose Table. In the $x_1$ column, enter the four $x$-values from the problem: $-1$, $0$, $1$, and $2$.

For each function, change the $y_1$ column to that function (for example, type y1 = (1/25)^x1, or use the function labels from your graph). Compare the Desmos $y$-values at $x=-1,0,1,2$ to the problem's table. The function whose values match all four table entries is the correct choice.

From the table:

• $g(-1)=25$
• $g(0)=1$
• $g(1)=\dfrac{1}{25}$
• $g(2)=\dfrac{1}{625}$

Check how $g(x)$ changes as $x$ increases by 1:

• From $x=-1$ to $x=0$: $25 \to 1$ (divide by 25, or multiply by $\dfrac{1}{25}$)
• From $x=0$ to $x=1$: $1 \to \dfrac{1}{25}$ (again multiply by $\dfrac{1}{25}$)
• From $x=1$ to $x=2$: $\dfrac{1}{25} \to \dfrac{1}{625}$ (again multiply by $\dfrac{1}{25}$)

So each time $x$ goes up by 1, $g(x)$ is multiplied by $\dfrac{1}{25}$, which is exactly the behavior of an exponential function with base $\dfrac{1}{25}$.

Notice that every value of $g(x)$ in the table is positive.

Now look at the answer choices with a negative sign in front:

• $g(x) = -25^x$ means $g(x)=-(25^x)$, which is always negative.
• $g(x) = -(1/25)^x$ means $g(x)=-(1/25)^x$, which is also always negative.

Both of these would give $g(0)=-1$, not $1$, and negative outputs for other $x$ values. So these two options cannot match the table and can be eliminated.

The remaining choices are the ones without a negative sign in front:

• One has base $25$.
• The other has base $\dfrac{1}{25}$.

Check which base fits the pattern you found in the table. In the table, as $x$ increases, $g(x)$ gets smaller by a factor of $\dfrac{1}{25}$ each time. A base larger than 1 (like $25$) would make the function values grow, not shrink, as $x$ increases. A base between 0 and 1 (like $\dfrac{1}{25}$) makes the function values shrink each time you increase $x$ by 1.

To be sure, test $x=1$ for each of the two remaining choices:

• With base $25$: $g(1)$ would be $25$, but the table says $g(1)=\dfrac{1}{25}$.
• With base $\dfrac{1}{25}$: $g(1)$ would be $\dfrac{1}{25}$, which matches the table.

Also, with base $\dfrac{1}{25}$, a negative exponent gives $g(-1)=25$, matching the table as well.

Therefore, the equation that defines $g$ is