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

Choice A is incorrect. This equation defines a function whose graph has a slope of $negative 39$ , 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 $left parenthesis 0 comma negative 39 right parenthesis$ , not $left parenthesis 0 comma 0 right parenthesis$ .