For the linear function f , the graph of y = f ( x ) in

Choice D is correct. An equation defining a linear function can be written in the form $f\left(x\right)=mx+b$, where $m$ is the slope and $\left(0,b\right)$ is the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane. It’s given that the graph of $y=f\left(x\right)$ has a slope of $39$, so $m=39$. It’s also given that the graph of $y=f\left(x\right)$ passes through the point $\left(0,0\right)$, so $b=0$. Substituting $39$ for $m$ and $0$ for $b$ in $f\left(x\right)=mx+b$ yields $f\left(x\right)=39x+0$, or $f\left(x\right)=39x$. Thus, the equation that defines $f$ is $f\left(x\right)=39x$.

Choice A is incorrect. This equation defines a function whose graph has a slope of $-39$, not $39$.

Choice B is incorrect. This equation defines a function whose graph has a slope of $\frac{1}{39}$, not $39$.

Choice C is incorrect. This equation defines a function whose graph has a slope of $1$, not $39$, and passes through the point $\left(0,-39\right)$, not $\left(0,0\right)$.