Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find functions \(f\) and \(g\) such that \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function, i.e., \(f(x) \neq x\) and \(g(x) \neq x .\) Answers to these problems are not unique. \(h(x)=\sqrt{5 x^{2}+3}\)

Step-by-Step Solution

Answer

The functions \(f\) and \(g\) that satisfy \(h(x)=f(g(x))\) and neither \(f\) nor \(g\) is the identity function are \(f(x) = \sqrt{x}\) and \(g(x) = 5x^{2}+3, respectively.

Step 1: Identification of Function g

We start by identifying function \(g\). Looking at \(h(x)=\sqrt{5 x^{2}+3}\), we can say that \(g(x)\) is the function under the square root. Hence, we have \(g(x)=5x^{2}+3\). We confirm that this is not an identity function since for an identity function, the output should equal the input, which is not the case here.

Step 2: Identification of Function f

Next, we identify function \(f\). Given the form of \(h(x)\), it seems that \(f(x)\) is the square root function. Hence, we can define \(f(x)=\sqrt{x}\). This function is also not an identity function, because an identity function should satisfy \(f(x)=x\) for all x.

Step 3: Verification

Finally, we verify that our chosen functions \(f\) and \(g\) satisfy the condition given in the problem, i.e., \(h(x)=f(g(x))\). Replacing \(f\) and \(g\) with their corresponding definitions, we have \(f(g(x)) = \sqrt{5x^{2}+3}\). Therefore, our chosen functions \(f\) and \(g\) are correct

Key Concepts

Function Composition

Function composition is akin to a culinary recipe where combining ingredients in sequence creates a unique dish. In mathematics, composite functions blend two functions together, one following another. Instead of preparing food, we're evaluating numerical outcomes. For a compelling composition, we blend function 'g' with 'f' to cook up a new mathematical expression, written as \( f(g(x)) \). However, unlike a chef who can toss ingredients at will, we must meticulously follow the original recipe: function 'g' is evaluated first, followed by feeding its outcome to function 'f'. The intriguing part is that we can often create various function pairs that result in the same final dish, or in our case, mathematical expression.

Square Root Function

The square root function, resembling a chef's precise technique, extracts the root of its ingredients. It's represented by \( f(x) = \sqrt{x} \), like a culinary technique that consistently transforms its ingredients, the square root function consistently processes numbers. It takes a number 'x' and produces its square root, revealing the number that, when multiplied by itself, equals 'x'. This function is fundamental in numerous recipes throughout the mathematical cookbook, notably in quadratic equations and Pythagorean delights. This is not an identity function as it morphs the input rather than merely presenting it as is.

Identity Function

On the flip side, the identity function is the culinary equivalent of serving raw, unadulterated ingredients in their natural state. Described by \( f(x) = x \), it's the mathematical reflection in the mirror, showing the input number exactly as it is, with no transformations or modifications. This is akin to a chef presenting a tomato in its pure form, highlighting its innate qualities without alteration. The identity function is a critical concept in understanding transformations in mathematics, as it represents the baseline, the untouched starting point from which other functions deviate and add their unique flavors.