Question

Tangent Lines Find equations of the tangent lines to the graph of \(f(x)=x /(x-1)\) that pass through the point \((-1,5) .\) Then graph the function and the tangent lines.

Short answer

The equations of the tangent lines to the function \(f(x) = x / (x - 1)\) that pass through the point (-1,5) are \(y = -4x - 3\) and \(y = 2x + 7\).

Step-by-step solution

Derivative of the Given Function

Find the derivative of the function \(f(x) = x / (x - 1)\). By applying the quotient rule, \(f'(x) = (1*(x - 1) - x*1) / (x - 1)^2 = 1 / (x - 1)^2.\)

Finding the x-values of the Tangent Points

Using the derivative, you can find the x-values where the tangent lines pass through the point (-1,5). Since a tangent line should have the same slope as the function at that point, and this line also goes through (-1,5), the slope m would be (f(x) - 5) / (x - (-1)) = 1 / (x - 1)^2. After simplifying and solving for x, two solutions can be found, x = 0 and x = -2.

Finding the Equation of the Tangent Lines

Now that the x values of the tangent points are known (x = 0, x = -2), the equations of the tangent lines can be determined by using the point-slope form y - y1 = m(x - x1), where (x1, y1) can be either (0, f(0)) or (-2, f(-2)), and m = 1 / (x - 1)^2.

Graphing the Function and the Tangent Lines

After obtaining the equations of the tangent lines, these lines can be plotted on the same set of axes as the function \(f(x) = x / (x - 1)\) to visually verify the solution. The function should appear as a hyperbola, while the tangent lines should intersect this curve at the points determined in step 2, and also pass through the point (-1,5)