In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
e. Plot the functions f and g, the identity, the two tangent lines, and the line segment joining the points (x_0, f(x_0)) and (f(x_0), x_0). Discuss the symmetries you see across the main diagonal y=x.


72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Verified step by step guidance

1. Identify the given function: \(f(x) = 2 - x - x^{3}\), with the domain \(-2 \leq x \leq 2\), and the point of interest \(x_0 = \frac{3}{2}\).
2. Calculate the value of the function at \(x_0\): compute \(f\left(\frac{3}{2}\right)\) to find the point \((x_0, f(x_0))\) on the graph of \(f\).
3. Find the inverse function \(g = f^{-1}\) implicitly or using a CAS, and evaluate \(g\) at \(f(x_0)\) to get the point \((f(x_0), x_0)\) on the graph of \(g\).
4. Determine the derivatives \(f'(x)\) and \(g'(x)\) at the points \(x_0\) and \(f(x_0)\) respectively, to find the slopes of the tangent lines to \(f\) and \(g\) at these points.
5. Plot the following on the same coordinate system: the function \(f\), its inverse \(g\), the identity line \(y = x\), the tangent line to \(f\) at \((x_0, f(x_0))\), the tangent line to \(g\) at \((f(x_0), x_0)\), and the line segment joining the points \((x_0, f(x_0))\) and \((f(x_0), x_0)\). Observe and discuss the symmetry of \(f\) and \(g\) about the line \(y = x\).

Key Concepts

Inverse Functions and Their Graphical Relationship

An inverse function reverses the roles of inputs and outputs of the original function, such that if y = f(x), then x = f⁻¹(y). Graphically, the function and its inverse are symmetric about the line y = x, meaning each point (a, b) on f corresponds to (b, a) on f⁻¹. Understanding this symmetry is key to analyzing the plots and their relationships.

Derivatives and Tangent Lines

The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point. For inverse functions, the derivative at corresponding points are reciprocals, reflecting the slope relationship of their tangent lines. Tangent line approximations help visualize local linear behavior and are essential for comparing f and its inverse.

Symmetry About the Line y = x

The line y = x acts as a mirror for a function and its inverse, meaning their graphs are reflections of each other across this line. This symmetry extends to tangent lines and points, where the segment joining (x₀, f(x₀)) and (f(x₀), x₀) lies on this diagonal. Recognizing this symmetry aids in understanding the geometric relationships in the problem.