Problem 57 - Calculus Single Variable

Question

An affine function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has the form \(f(x)=a x+b\), where \(a\) and \(b\) are constants. Prove that the composition of two affine functions is affine and that the inverse of an invertible affine function is affine.

Step-by-Step Solution

Answer

The composition and inverse of affine functions are both affine.

Step 1: Understanding the Composition of Affine Functions

Two affine functions can be described by their forms: suppose \(f(x) = a_1x + b_1\) and \(g(x) = a_2x + b_2\). We will find the composition \(h(x) = f(g(x))\).

Step 2: Calculating the Composition

Substitute the expression for \(g(x)\) into \(f(x)\):\[ h(x) = f(g(x)) = f(a_2x + b_2) = a_1(a_2x + b_2) + b_1 = a_1a_2x + a_1b_2 + b_1.\]

Step 3: Verifying the Composition is Affine

The resulting expression \(h(x) = a_1a_2x + (a_1b_2 + b_1)\) is of the form \(ax + b\), proving it is also an affine function.

Step 4: Defining the Inverse of an Affine Function

To find the inverse of an affine function \(f(x) = ax + b\), assume \(f(x) = y\) and solve for \(x\):\[ y = ax + b \Rightarrow x = \frac{y - b}{a}.\]

Step 5: Expressing the Inverse Function

The inverse function can be expressed as \(f^{-1}(y) = \frac{y - b}{a}\). This simplifies to \(f^{-1}(x) = \frac{1}{a}x - \frac{b}{a}\).

Step 6: Confirming the Inverse is Affine

The expression \(f^{-1}(x) = \frac{1}{a}x - \frac{b}{a}\) is clearly of the form \(ax + b\), confirming that the inverse is also an affine function.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics. It involves applying one function to the results of another function. Imagine you have two boxes, each with different inputs and outputs. You put something into the first box, get a result, and then put that result into the second box.

This is essentially what function composition is. For example, if you have two affine functions, such as \(f(x) = a_1x + b_1\) and \(g(x) = a_2x + b_2\), the composition \(h(x) = f(g(x))\) shows the result of sequential operations.

• Start by inserting \(x\) into \(g\), so it becomes \(g(x) = a_2x + b_2\).
• Then, this output serves as the input for \(f\), giving us \(f(g(x)) = a_1(a_2x + b_2) + b_1\).

The result after all steps is another function of the form, showing that the composition of affine functions remains an affine function.

Inverse Function

The inverse function is like a magical undo button. If a function transforms an input \(x\) to an output \(y\), the inverse function takes \(y\) back to \(x\). In simpler terms, it reverses what the function does.

For example, let's say we have an affine function \(f(x) = ax + b\). To uncover its inverse, we swap the roles of \(x\) and \(y\), solve the equation \(y = ax + b\) for \(x\), and find that the inverse function is \(f^{-1}(x) = \frac{1}{a}x - \frac{b}{a}\).

This means that if \(f\) multiplied and added, \(f^{-1}\) will divide and subtract to get back to the original value. Thus, this inverse also maintains the affine form, proving that even when the process is reversed, it stays within the affine family.

Linear Functions

Linear functions are a special type of function simplified by their straightforward form. They typically follow the structure \(f(x) = mx + c\), where the graph is always a straight line.

Affine functions are an expanded version of this, where they can include constant shifts or translations but maintain the essence of linear transformations.

• A linear function is entirely dependent on its slope, \(m\), which tells you how steep or flat the line is.
• For a linear function, transformations involve shifting or stretching, staying true to a 'line' on a graph but possibly with vertical adjustments thanks to \(c\).

Every affine function can be considered an honorary linear function, emphasizing the significant role that linear functions play in developing mathematical understanding of various function types.

Mathematical Proof

Mathematical proof is the convincing and logical argument given to demonstrate a mathematical statement is true. In building these arguments, mathematicians use well-known and established concepts, like axioms and previously proven theorems.

To prove that a composition or an inverse of an affine function remains affine, the solution involves showing step-by-step, using the logic and structures of known forms.

• For the composition, by expanding \(f(g(x))\), showing the resulting expression has the affine form, is a straightforward demonstration of the theorem.
• For the inverse, we rearrange \(y = ax + b\) to form \(x\), demonstrating that you will arrive at a function \(f^{-1}(x) = \frac{1}{a}x - \frac{b}{a}\) which fits the affine form.

Through these processes, the consistency of the affine nature is preserved, providing a clear and concise understanding of the concepts through mathematical proof.