x g x - 1 25 0 1 1 1 25 2 1 625 For the

Choice D is correct. It's given that function $g$ is exponential. Therefore, an equation defining $g$ can be written in the form $g\left(x\right)=a{\left(b\right)}^x$, where $a$ and $b$ are constants. The table shows that when $x=0$, $g\left(x\right)=1$. Substituting $0$ for $x$ and $1$ for $g\left(x\right)$ in the equation $g\left(x\right)=a{\left(b\right)}^x$ yields $1=a{\left(b\right)}^0$, which is equivalent to $1=a$. Substituting $1$ for $a$ in the equation $g\left(x\right)=a{\left(b\right)}^x$ yields $g\left(x\right)={\left(b\right)}^x$. The table also shows that when $x=1$, $g\left(x\right)=\frac{1}{25}$. Substituting $1$ for $x$ and $\frac{1}{25}$ for $g\left(x\right)$ in the equation $g\left(x\right)={\left(b\right)}^x$ yields $\frac{1}{25}={\left(b\right)}^1$, which is equivalent to $\frac{1}{25}=b$. Substituting $\frac{1}{25}$ for $b$ in the equation $g\left(x\right)={\left(b\right)}^x$ yields $g\left(x\right)={\left(\frac{1}{25}\right)}^x$.

Choice A is incorrect. For this function, $g\left(1\right)$ is equal to $-25$, not $\frac{1}{25}$.

Choice B is incorrect. For this function, $g\left(1\right)$ is equal to $-\frac{1}{25}$, not $\frac{1}{25}$.

Choice C is incorrect. For this function, $g\left(1\right)$ is equal to $25$, not $\frac{1}{25}$.