Problem 9 - Algebra Form and Function

Question

Express the function as a composition of two simpler functions. $$ y=\sqrt{x^{2}+1} $$

Step-by-Step Solution

Answer

Question: Express the function $$y=\sqrt{x^2+1}$$ as a composition of two simpler functions. Answer: The given function $$y=\sqrt{x^2+1}$$ can be expressed as a composition of two simpler functions $$f(x)=x^2+1$$ and $$g(x)=\sqrt{x}$$, such that $$y=g(f(x))=\sqrt{x^2+1}$$.

Step 1: Identify the two simpler functions

The given function is $$y=\sqrt{x^{2}+1}$$. In order to express this as a composition of two simpler functions, we can consider the following two functions:$$ f(x)=x^2+1 \\ g(x)=\sqrt{x} $$ Notice how these two functions are simpler than the given function and it appears that their composition might result in the original function. Let's proceed to confirm this.

Step 2: Find the composition of the two functions

Now we will find the composition of these two functions. We are aiming to find the composition $$g(f(x))$$ such that it results in the original function $$y=\sqrt{x^{2}+1}$$. Let's find out if these two functions can form such a composition: $$ g(f(x)) = g(x^2+1) $$ Since the function $$g(x)=\sqrt{x}$$, we just need to substitute $$x^2+1$$ into $$g(x)$$ to obtain the composition: $$ g(f(x)) = \sqrt{x^2+1} $$

Step 3: Verify the composition

The composition of our two simpler functions, $$g(f(x))$$, is indeed$$ g(f(x))=\sqrt{x^{2}+1} $$ This is the same as the original function $$y=\sqrt{x^{2}+1}$$, which confirms that the composition of the simpler functions, $$f(x)$$ and $$g(x)$$, results in the given function $$y$$.

Key Concepts

Square Root Function

The square root function is often denoted as \(g(x) = \sqrt{x}\). It is the inverse of squaring a number. The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). For example, the square root of 9 is 3, because \(3 \times 3 = 9\). The domain of the square root function is restricted to non-negative numbers because the square root of a negative number is not defined within the set of real numbers. The range of this function is also non-negative, meaning it only outputs non-negative numbers.

• Special Property: The function always returns the non-negative root, known as the principal square root.
• Graph Characteristics: The graph of \(g(x) = \sqrt{x}\) is a gentle curve rising from the origin, moving rightward along positive \(x\)-values.

Understanding this function allows it to be used in composition with other functions, such as a quadratic function to create a composition like \(\sqrt{x^2+1}\).

Quadratic Function

A quadratic function is expressed in the standard form \(f(x) = ax^2 + bx + c\). In this exercise, we considered the simpler quadratic function \(f(x) = x^2+1\). The general form uses a squared term as its highest degree, making it a second-degree polynomial function.

• Nature of Quadratics: These functions often produce parabolic graphs. The direction of the parabola (opening upwards or downwards) depends on the sign of the coefficient \(a\). For instance, if \(a > 0\), the parabola opens upwards.
• Vertex: The point where the parabola changes direction is its vertex, and it can either be a maximum or a minimum point.

This function is used in the given composition to form part of the expression under the square root, resulting in \(\sqrt{x^2+1}\). Its usage in compositions aligns with many real-world applications, like physics and engineering, where quadratic relationships are common.

Mathematical Notation

Mathematical notation is a set of symbols and conventions used to communicate mathematical ideas clearly and succinctly. In the exercise, we use various notations:

• Function Notation: Functions are written as \(f(x)\), which represents the output of the function \(f\) for an input \(x\).
• Composition of Functions: Notated as \(g(f(x))\), this represents the application of function \(g\) to the result of function \(f\) applied to \(x\).
• Square Root: Denoted by \(\sqrt{x}\), indicating the principal square root or the non-negative root.

These notations simplify complex expressions and allow us to write concise mathematical statements. Understanding these notations is crucial in comprehending and manipulating functions, as in this exercise where we expressed \(y = \sqrt{x^2+1}\) using simpler functions.