Question

Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(F(x)=2 x+8\). Let \(G: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(G(y)=\) \((y-8) / 2\). Prove that \(F \circ G=I d_{\mathrm{R}}\) and \(G \circ F=I d_{\mathrm{R}}\).

Short answer

Both \(F \circ G = Id_R\) and \(G \circ F = Id_R\) are proven, meaning they are inverse functions.

Step-by-step solution

Understand the Problem

The exercise requires proving that both, the composition of functions \(F\) and \(G\) in one order and the reverse, are equal to the identity function. We will show that \((F \circ G)(x) = x\) and \((G \circ F)(y) = y\). This means applying one function, then the other, results in a net effect of zero, akin to the identity function.

Prove \(F \circ G = Id_R\)

Compute \(F(G(x))\) for an arbitrary \(x \in \mathbb{R}\). Begin by substituting \(G(x)\) into \(F\), so \(G(x) = (x-8)/2\). Now apply \(F\): \[ F(G(x)) = F\left(\frac{x-8}{2}\right) = 2\left(\frac{x-8}{2}\right) + 8. \]Simplify this expression:\[ 2\left(\frac{x-8}{2}\right) + 8 = x-8 + 8 = x. \]Thus, \(F(G(x)) = x\), showing \(F \circ G = Id_R\).

Prove \(G \circ F = Id_R\)

Compute \(G(F(y))\) for an arbitrary \(y \in \mathbb{R}\). Start by substituting \(F(y)\) into \(G\), so \(F(y) = 2y + 8\). Now apply \(G\): \[ G(F(y)) = G(2y + 8) = \frac{(2y + 8) - 8}{2}. \]Simplify this expression:\[ \frac{2y + 8 - 8}{2} = \frac{2y}{2} = y. \]Thus, \(G(F(y)) = y\), proving \(G \circ F = Id_R\).

Conclusion: Verify the Identity Function

Both compositions \(F \circ G\) and \(G \circ F\) result in the identity function, meaning \(F(G(x)) = x\) and \(G(F(y)) = y\). Since in both cases the output equals the input, we prove the functions behave as inverses.

Key concepts

Inverse Functions

Inverse functions play a key role in mathematics by reversing the effect of another function. In the context of the given functions, the function \(F(x) = 2x + 8\) and \(G(y) = (y-8)/2\) are inverses of each other. This means that applying \(F\) and then \(G\) (or vice versa) results in the original input value. This characteristic is the fundamental property of inverse functions.

• To find an inverse function, you essentially solve the equation for the opposite variable.
• For example, to get \(G\) from \(F\), you set \(y = F(x)\) and solve for \(x\), leading you to \(G\).

Inverse functions are symbolically written as \(G = F^{-1}\), indicating that \(G\) reverses the effect of \(F\). When two functions are inverses, their composition yields the identity function, showing that the operations cancel each other out and leave you with the input as the output.

Identity Function

An identity function is one of the simplest yet essential types of functions in mathematics. It is a function that always returns the same value that was used as its input. For a set of real numbers \(\mathbb{R}\), the identity function is represented as \(Id_R(x) = x\).
In the exercise, you are asked to demonstrate that the compositions \(F \circ G\) and \(G \circ F\) result in the identity function. This assures you that applying the function and its inverse are essentially trivial actions: they take you full circle back to your starting point, much like how reversing a journey leads you back home.

• The identity function acts like a "do-nothing" operation. Whatever you input is exactly what you get out.
• As shown in the solutions, \((F \circ G)(x) = x\) and \((G \circ F)(y) = y\), confirming that each returns the original input value.

Understanding the identity function helps grasp the concept of inverses, as achieving the identity function through composition is a confirmation that two functions are true inverses.

Real-valued Functions

Real-valued functions are an integral part of calculus and higher mathematics. These functions have real numbers as their input and output, making them tremendously valuable for modeling and solving real-world problems.For example, in the exercise problem, functions \(F\) and \(G\) both map from the real numbers \(\mathbb{R}\) to the real numbers \(\mathbb{R}\).
This means every input is a real number and the functions produce real number outputs which are both applicable and easy to work with in various contexts.

• This classification makes sure the function's operations (like addition, multiplication, etc.) are well defined over the real numbers.
• Real-valued functions are essential in continuous mathematics and serve as a fundamental concept when dealing with calculus and analysis.

By working with real-valued functions, you can efficiently describe continuous processes, movements, and changes observed in nature and various fields of engineering and science. Real-valued functions paired with proper operations like inversion and composition allow complex operations to be checked or even simplified.