Question

Tangent Lines Find equations of the tangent lines to the graph of \(f(x)=(x+1) /(x-1)\) that are parallel to the line \(2 y+x=6 .\) Then graph the function and the tangent lines.

Short answer

The equations of the tangent lines to the function \(f(x)=(x+1)/(x-1)\) that are parallel to the line \(2y+x=6\) are obtained after following through the steps. After plotting, the function and the two tangent lines should intersect at two points.

Step-by-step solution

Determine the Slope of the Given Line

First, write the equation of the line in slope-intercept form \(y = mx+b\), where m is the slope. To do this, we change \(2y + x = 6\) to \(y = -0.5x + 3\). So the slope of the given line is -0.5.

Find the Derivative of the Function

The derivative of \(f(x)=(x+1)/(x-1)\) can be found using the quotient rule. The derivative \(f'(x) = (x-1)^{-2}\). The slope of the tangent line to the function at any point is given by the derivative's value at that point.

Set the Derivative Equal to the Line's Slope

To find the x-values where the function has tangent lines parallel to the given line, set the derivative equal to the slope of the line, such that \((x-1)^{-2}=-0.5\). This gives two solutions.

Solve for X-Values

Solving \((x-1)^{-2}=-0.5\) yields two x-values, meaning there are two points on \(f(x)\) where the tangent lines have a slope of -0.5.

Find the Y-Values

Substitute the x-values from the previous step into the original function \(f(x)\) to find the corresponding y-values.

Write Equations of the Tangent Lines

Using the slope from Step 1 and the points from Step 5, write the equations of the tangent lines in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.

Graph of the Function and Tangent Lines

Using a graphing software or paper, graph the function and the tangent lines obtained from the step above.