Problem 55 - Precalculus with Limits

Question

In Exercises 49-62, (a) find the inverse function of \(f\) (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). \(f{x} = \frac{4}{x}\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = \frac{4}{x}\) is \(f^{-1}(x) = \frac{4}{x}\). Both graphs overlap, meaning they are the same function. For both, the domain is \(x ∈ ℝ, x ≠ 0\) and the range is \(y ∈ ℝ, y ≠ 0\)

Step 1: Finding the Inverse Function

The inverse of a function can be found by switching the places of y and x. So, start with \(f(x) = \frac{4}{x}\) and switch y and x to get \(x = \frac{4}{y}\). Solving for y will give us the inverse function. Therefore, by multiplying both sides of the equation by y and then dividing by x we get the inverse function \(f^{-1}(x) = \frac{4}{x} \)

Step 2: Graphing the Function and its Inverse

The graph of the function \(f(x) = \frac{4}{x}\) is a rectangular hyperbola with a center at the origin. The graph of its inverse, \(f^{-1}(x) = \frac{4}{x}\), is exactly the same because the function and its inverse are symmetrical with respect to the line \(y = x\). This is a special case that happens when the function \(f\) is its own inverse.

Step 3: Describing the Relationship Between the Graphs of the Function and its Inverse

In general, a function and its inverse are symmetrical along the line \(y = x\). In this special case, the function \(f(x) = \frac{4}{x}\) and its inverse \(f^{-1}(x) = \frac{4}{x}\) overlap perfectly on the graph, meaning they are the same function.

Step 4: Stating the Domain and Range of the Function and its Inverse

The function \(f(x) = \frac{4}{x}\) is defined for \(x ≠ 0\). Therefore, the domain of \(f\) is \(x ∈ ℝ, x ≠ 0\), and since it can output any real number, the range of \(f\) is also \(y ∈ ℝ, y ≠ 0\). Similarly, the domain and range of its inverse \(f^{-1}(x)\), are both \(x, y ∈ ℝ, x, y ≠ 0\) because, in this case, the function and its inverse are the same.

Key Concepts

Graphing Inverse Functions

Graphing the inverse of a function involves a distinct property: symmetry about the line \( y = x \). To visualize this, you can start with the function's graph and 'flip' it over this line. For instance, if the original function has a point at (2, 3), the inverse function will have a point at (3, 2). The procedure is rather straightforward if the algebraic form of the function is known: solve the equation for the inverse function, then plot its graph.

When graphing inverses, key features to observe include intersections with axes, asymptotes, and intervals where the function increases or decreases. These features are mirrored for the inverse function, helping to establish an accurate graph. It's vital to note that not all functions have inverses that are also functions, a concept known as 'function inverses' which must pass the horizontal line test.

Symmetry in Inverse Functions

Symmetry takes a pivotal role when studying inverse functions. The symmetry is not just a visual perk but an inherent mathematical property. By trading every \(x\) with \(y\) in the equation of a function, you delineate its inverse. This switch manifests graphically as reflection across the line \(y = x\).

For the rectangular hyperbola \(f(x) = \frac{4}{x}\), which is unique in being its own inverse, this symmetry becomes self-identity. Both the hyperbola and its inverse fall on the same curve, splitting the coordinate plane into four symmetrical regions. Recognizing this allows students to predict and check their work when finding and graphing inverse functions.

Domain and Range of Inverse Functions

Understanding the domain and range of inverse functions is about appreciating the role swap between outputs and inputs. For a function \(f\) with domain \(D\) and range \(R\), its inverse \(f^{-1}\) will have domain \(R\) and range \(D\). In the context of our original function \(f(x) = \frac{4}{x}\), the domain or the set of all possible inputs excludes zero, as division by zero is undefined. Consequently, the range or all possible output values also excludes zero, since the hyperbola never meets the axes.

In the special case of an invertible function that serves as its own inverse, both the domain and range reflect this self-contained symmetry. It is crucial to ensure that for any values considered, one must adhere to the restrictions set by non-permissible values such as division by zero.

Rectangular Hyperbola

A rectangular hyperbola is marked by its characteristic shape: each branch bends away from two perpendicular asymptotes, creating a 'rectangular' window. In algebraic terms, a common form for such a curve is \(xy = c\) where \(c\) is a constant. When you're dealing with the function \(f(x) = \frac{4}{x}\), its graph follows this classic hyperbolic pattern with \(c = 4\).

The graph is situated in two of the four quadrants of the coordinate plane, due to the undefined nature of the function at zero. As students graph more functions like this, they'll learn to quickly identify the hallmarks of a rectangular hyperbola—its symmetry axes, where the graph approaches but never crosses, and the direction it follows as the absolute values of \(x\) and \(y\) increase.