Problem 7 - Calculus of a Single Variable

Question

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=\frac{1}{x}, \quad \quad \quad g(x)=\frac{1}{x}\)

Step-by-Step Solution

Answer

Analytically when we substitute \(g(x)\) into \(f(x)\) and vice versa, we got \(x\), which proves \(f(x)\) and \(g(x)\) are inverse functions. And the graphs of \(f(x)\) and \(g(x)\) are reflection over the line \(y = x\), which confirms they are inverse functions graphically as well.

Step 1: Test for inverse properties (analytical)

To prove \(f\) and \(g\) are inverse functions, substitute \(g(x)\) into \(f(x)\) and \(f(x)\) into \(g(x)\) and see if the composition equals \(x\). 1. Substituting \(g(x)\) into \(f(x)\), we obtain \(f(g(x))=f\left(\frac{1}{x}\right) = \frac{1}{1/x} = x\).2. Substituting \(f(x)\) into \(g(x)\), we obtain \(g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{1/x} = x\).So, both \(f(g(x))\) and \(g(f(x))\) equal \(x\), confirming \(f\) and \(g\) are inverses analytically.

Step 2: Test for inverse properties (graphical)

The graphs of \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x}\) are both hyperbolas reflecting over the origin. The graphs of two inverse functions should be reflected over the line \(y=x\). Since the line \(y=x\) passes through the origin in the same way that the graphs \(f(x)\) and \(g(x)\) do, they are a reflection of each other over the line \(y=x\). This suggests that they are likely inverse functions. Hence, both analytically and graphically, \(f(x)\) and \(g(x)\) are inverses of each other.

Key Concepts

Analytical Methods

To verify that two functions such as \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x} \) are inverses, analytical methods are used, where we focus on their composition. This means performing substitutions to check certain properties. Here's how it works:

• Compose \( f \) with \( g \): This involves substituting \( g(x) \) into \( f(x) \). We calculate \( f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{1/x} = x \). If the result is \( x \), this supports that they may be inverses.


• Compose \( g \) with \( f \): In this step, \( f(x) \) is substituted into \( g(x) \). We find that \( g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{1/x} = x \). Again, if this simplifies to \( x \), it further confirms their inverse relationship.

If both compositions yield \( x \), the functions are indeed inverses of each other. This method relies on algebraic manipulation, providing a concrete form of proof without graphing.

Graphical Methods

Graphically verifying inverse functions involves examining their graphs for a specific symmetrical property. When \( f(x) \) and \( g(x) \) are inverses, their graphs will be reflections over the line \( y = x \). This property helps us visualize the inverse relationship.
Consider the graph of \( f(x) = \frac{1}{x} \). It is a hyperbola, and the same can be said for \( g(x) = \frac{1}{x} \), since these functions are identical and thus overlap perfectly.

When the functions are inverses, the reflection symmetry over the line \(y = x\) should be observed clearly:

• The line \( y = x \) serves as a mirror. If one function can be 'flipped' over this line to exactly match the other, they are inverses.


• In this exercise, since the graphs of \( f \) and \( g \) are indeed symmetric with respect to \( y = x \), graphical verification suggests they are inverses.

Graphical methods enhance understanding by providing a visual validation, complementing the analytical approach.

Composition of Functions

The concept of the composition of functions is integral when proving two functions are inverses. It involves combining two functions to produce a new function by substituting one into the other.
Specifically, for functions \( f \) and \( g \), their compositions, \( f(g(x)) \) and \( g(f(x)) \), should both simplify to \( x \) if they are true inverses.

Here's a breakdown:

• Definition of Composition: Composition \( f(g(x)) \) means that you apply \( g(x) \) first and then apply \( f \), and vice versa for \( g(f(x)) \).


• Verifying Inverses: For functions to be inverses:
  • \( f(g(x)) = x \)
  • \( g(f(x)) = x \)
  These both must be satisfied.

The composition of functions acts like a bridge showing how tightly coupled the operations of \( f \) and \( g \) are, each running "backward" to undo the other's effect, returning you to the original input \( x \). This analytical tool solidifies understanding of inverse relationships.