Final answer:
The equation of the tangent line to the graph of the function at x=0 is y = 14x - 9. The slope of the tangent line is determined by differentiating the function and evaluating at x=0.
Step-by-step explanation:
The student is asking for the equation of the tangent line to the graph of a function at a specific point where x=0. To find this equation, we first need to find the slope of the tangent line by differentiating the function f(x). Then, we'll evaluate this derivative at x=0 to find the slope of the tangent line at that point. Finally, we'll use the point-slope form of the line equation to create the tangent line's equation.
To find the derivative of f(x) = (5x - 9) / (x + 1), we can use the quotient rule:
f'(x) = [(x+1)(5) - (5x-9)(1)] / (x+1)^2.
Simplifying, we get f'(x) = (5x + 5 - 5x + 9) / (x+1)^2 = 14 / (x+1)^2.
Evaluating this at x=0 gives us f'(0) = 14.
The y-intercept of the function when x=0 is f(0) = -9.
Now, using the point-slope form y - y1 = m(x - x1), with m = 14, x1 = 0, and y1 = -9, the tangent line equation is y + 9 = 14(x - 0), or y = 14x - 9.