Problem 92 - Algebra and Trigonometry

Question

The Identity Function The function \(I(x)=x\) is called the identity function. Show that for any function \(f\) we have \(f \circ I=f, I^{\circ} f=f,\) and \(f^{\circ} f^{-1}=f^{-1} \circ f=I .\) (This means that the identity function \(I\) behaves for functions and composition just the way the number 1 behaves for real numbers and multiplication.)

Step-by-Step Solution

Answer

The identity function \(I(x) = x\) behaves neutrally in composition: \(f \circ I = f\), \(I \circ f = f\), and \(f \circ f^{-1} = f^{-1} \circ f = I\).

Step 1: Define Function Composition

The composition of two functions, denoted as \((f\circ g)(x)\), is defined by applying the functions in sequence: \(f(g(x))\). This means you first apply the function \(g\) to the input \(x\), and then apply \(f\) to the result.

Step 2: Verify \(f \, \circ \, I = f\)

For any function \(f\), when composed with the identity function \(I(x) = x\), we have:\[(f \circ I)(x) = f(I(x)) = f(x)\] Thus, the composition \(f \circ I\) results in the function \(f\) itself.

Step 3: Verify \(I \, \circ \, f = f\)

Similarly, we can find the composition of the identity function with any function \(f\):\[(I \circ f)(x) = I(f(x)) = f(x)\]Here, \(I\) returns its input as output, so the composition results in \(f\).

Step 4: Define Inverse Function

An inverse function \(f^{-1}\) reverses the mapping of \(f\), so for every \(y\) in the range of \(f\), \(f^{-1}(f(y)) = y\). For the pair of functions \(f^{-1} \) and \(f \), the composition will yield an identity function.

Step 5: Verify \(f^{\circ} f^{-1} = I\)

To check \(f^{\circ} f^{-1} = I\):\[(f \circ f^{-1})(x) = f(f^{-1}(x)) = x = I(x)\] This composition acts as the identity function \(I(x) = x\).

Step 6: Verify \(f^{-1} \circ f = I\)

To check \(f^{-1}\circ f = I\):\[(f^{-1} \circ f)(x) = f^{-1}(f(x)) = x = I(x)\]This results in the identity function \(I(x) = x\) as well, proving \(f^{-1}\circ f = I\).

Step 7: Final Conclusion

The identity function \(I(x) = x\) acts neutrally in function composition just like the number 1 does in multiplication, demonstrating that \(f \circ I = f\), \(I \circ f = f\), and for inverse functions, \(f^{\circ} f^{-1} = I\) and \(f^{-1} \circ f = I\).

Key Concepts

Function Composition

Function composition is a way of combining two functions to create a new function. It's denoted as \(f \circ g\), which means you apply the function \(g\) first and then apply \(f\) to the result. This is expressed mathematically as \(f(g(x))\). The beauty of function composition lies in its ability to synthesize complex functions from simpler ones. Think of it like building a machine: you pass the input through one part (a function), which processes it and sends it to another part for further processing.

The identity function \(I(x) = x\) plays a special role in this process. When you compose any function \(f\) with the identity function, either as \(f \circ I\) or \(I \circ f\), you get the original function \(f\) back. Mathematically, it looks like this:

• \(f \circ I(x) = f(I(x)) = f(x)\)
• \(I \circ f(x) = I(f(x)) = f(x)\)

These equations show that the identity function acts like a neutral element in the world of functions, similar to how multiplying by 1 leaves a number unchanged.

Inverse Function

An inverse function essentially undoes the action of the original function. If \(f\) maps an input \(x\) to \(y\), its inverse \(f^{-1}\) maps \(y\) back to \(x\). In notation, this relationship is written as \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(y)) = y\). The existence of an inverse function is dependent on the original function being a bijection, meaning it is both one-to-one and onto.

Composing a function with its inverse yields the identity function. It means applying \(f\) and then \(f^{-1}\) (or vice versa) brings you back to where you started:

• \(f \circ f^{-1}(x) = f(f^{-1}(x)) = I(x) = x\)
• \(f^{-1} \circ f(x) = f^{-1}(f(x)) = I(x) = x\)

These compositions confirm that \(f^{-1}\) and \(f\) are inverses by returning the input as output, effectively neutralizing each other.

Algebraic Identity

In mathematics, an algebraic identity is a statement that holds true for all possible values involved. It’s like a universal truth written in numbers or symbols. One of the simplest algebraic identities involves the identity function, \(I(x) = x\). This function behaves in functional composition like the number 1 in multiplication. So, just as \(a \times 1 = a\) for any number \(a\), the identity function satisfies \(f \circ I = f\) and \(I \circ f = f\) for any function \(f\).

Moreover, just as \(a \times a^{-1} = 1\) when \(a^{-1}\) is the multiplicative inverse, \(f \) composed with \(f^{-1}\) gives the identity function: \(f \circ f^{-1} = I\) and \(f^{-1} \circ f = I\). These identities underline the symmetry and balance properties that are at the heart of algebraic manipulations and transformations.