Problem 19 - Calculus for Biology and Medicine

Question

Suppose that \(f(x)=x^{2}, x \geq 0\), and \(g(x)=\sqrt{x}, x \geq 0 .\) Typically, \(f \circ g \neq g \circ f\), but this is an example in which the order of composition does not matter. Show that \(f \circ g=g \circ f\).

Step-by-Step Solution

Answer

The compositions \(f \circ g\) and \(g \circ f\) both equal \(x\) for \(x \geq 0\), thus showing they are equal.

Step 1: Understand Function Composition

In function composition, given two functions \(f\) and \(g\), \(f \circ g\) means that you apply \(g\) first and then apply \(f\) on the result. On the other hand, \(g \circ f\) means you apply \(f\) first and then \(g\). Our goal is to show that \(f \circ g(x) = g \circ f(x)\).

Step 2: Calculate \(f \circ g(x)\)

To find \(f \circ g(x)\), start with \(g(x)\):\[g(x) = \sqrt{x}\]Next, apply \(f\) to \(g(x)\):\[f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2 = x\]

Step 3: Calculate \(g \circ f(x)\)

Now, find \(g \circ f(x)\) by starting with \(f(x)\):\[f(x) = x^2\]Next, apply \(g\) to \(f(x)\):\[g(f(x)) = g(x^2) = \sqrt{x^2} = |x|\]Since we are given \(x \geq 0\), \(|x| = x\).

Step 4: Compare the Results

Finally, we compare both results:For \(f \circ g(x)\):\[f(g(x)) = x\]For \(g \circ f(x)\):\[g(f(x)) = x\]Since \(f(g(x)) = g(f(x)) = x\), this verifies that \(f \circ g = g \circ f\).

Key Concepts

Identity Function

An identity function, denoted as \( I(x) = x \), is a function that returns its input without any changes. It's one of the simplest functions in mathematics and serves as a foundation for understanding more complex functions. By definition, the identity function acts as a neutral element in the operation of function composition. When you compose any function \( f(x) \) with the identity function, either before or after \( f(x) \), the result remains unchanged.\

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• \( f(I(x)) = f(x) \)
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• \( I(f(x)) = f(x) \)
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\Thus, for any function \( f \), the identity function has no effect on the computation. This property is crucial in situations requiring function compositions that need to maintain their original outputs, enabling smoother transitions and simpler calculations.

Square Function

The square function is defined as \( f(x) = x^2 \). It is a fundamental mathematical function where each input value is multiplied by itself, resulting in a non-negative output. Understanding the behavior of the square function is essential because it often serves as a building block for many mathematical models and real-world applications.\

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• The output is always positive or zero for real numbers.
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• It is symmetric around the y-axis, which means \( f(x) = f(-x) \).
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• All values \( x \geq 0 \) will produce non-negative results, satisfying the condition in our exercise.
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\Square functions are extensively used in physics and engineering, especially in calculations involving areas and energy, making them a key concept in both theoretical and applied mathematics.

Square Root Function

The square root function is given by \( g(x) = \sqrt{x} \). It’s the inverse of the square function for non-negative values of \( x \). This function effectively "undoes" the effect of squaring a number. Understanding the square root function is vital because it helps solve equations that involve power and inverse operations.\

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• It only applies to non-negative numbers, as square roots of negative numbers aren't real in the standard real number system.
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• The function returns the principal (positive) square root of \( x \).
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• It is a non-decreasing function, meaning \( g(x_1) \leq g(x_2) \) whenever \( x_1 \leq x_2 \).
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\This function is crucial in many areas, including geometry for calculating lengths of sides, physics for determining rates and energies, and in finance, where it helps in calculating compound interest returns. By composing this function with others, such as the square function, we can further explore and understand the symmetry and properties of mathematical operations.