Problem 18 - Calculus Single Variable

Question

Find a function \(g\) such that \(h=g \circ f\) \(h(x)=2 x^{2}+x-\sqrt[6]{x}+1, f(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

The function \(g(y)\) is \(2y^4 + y^2 - y^{2/6} + 1\).

Step 1: Understand the Relationship between h, g, and f

To find a function \(g\) such that \(h = g \circ f\), we recognize that \(g(f(x)) = h(x)\). Since \(f(x) = \sqrt{x}\), it follows that \(g(\sqrt{x}) = h(x)\). Our goal is to express \(g(y)\) where \(y = \sqrt{x}\).

Step 2: Substitute f(x) into h(x)

Since \(f(x) = \sqrt{x}\), substitute \(y = \sqrt{x}\) into the expression for \(h(x)\): \[ h(y^2) = 2(y^2)^2 + y^2 - \sqrt[6]{y^2} + 1 \]

Step 3: Simplify h(y²) in terms of y

Simplify \(h(y^2)\) where \(y = \sqrt{x}\): \[ 2(y^2)^2 + y^2 - \sqrt[6]{y^2} + 1 = 2y^4 + y^2 - (y^2)^{1/6} + 1 \] This simplifies the expression of \(h\) using \(y\) to match the form \(g(y)\).

Step 4: Define g(y)

Now define \(g(y)\): \[ g(y) = 2y^4 + y^2 - y^{2/6} + 1 \] This is the function \(g\) which, when composed with \(f(x) = \sqrt{x}\), gives \(h(x)\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics where two functions are combined to form a new function. In simpler terms, function composition involves taking one function and plugging it into another. This results in a third function.
For any functions \( g \) and \( f \), their composition is denoted as \( g \circ f \), which means that you first apply \( f \) to an input, and then apply \( g \) to the output of \( f \).
In our exercise, we've been given two functions: \( h(x) = 2x^2 + x - \sqrt[6]{x} + 1 \) and \( f(x) = \sqrt{x} \). The task is to find a function \( g \) such that composing \( f \) with \( g \) results in \( h \). In other words, \( h = g \circ f \), implying \( g(f(x)) = h(x) \).
Thus, understanding the concept of function composition helps us break down complex functions into manageable parts for easier manipulation and problem-solving.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions to achieve a certain form. When dealing with composite functions, it's often necessary to manipulate expressions to isolate specific terms or to evaluate functions more easily.
In the given problem, by suggesting \( h = g \circ f \), we imply \( g(f(x)) = h(x) \). With \( f(x) = \sqrt{x} \), we substitute \( y = \sqrt{x} \) into \( h(x) \) and write \( h(y^2) = 2(y^2)^2 + y^2 - \sqrt[6]{y^2} + 1 \).
To simplify this expression, we recognize that constants and powers of \( y \) need to be managed correctly. Through algebraic manipulation, we simplify it to \( h(y^2) = 2y^4 + y^2 - y^{2/6} + 1 \), giving us the expression for \( g(y) = 2y^4 + y^2 - y^{2/6} + 1 \).

• This process allows us to construct \( g(y) \), giving insight into how smaller algebraic pieces fit together into a larger function.
• Algebraic manipulation involves carefully applying arithmetic operations and properties of exponents to transform and understand expressions better.

Mathematical Modeling

Mathematical modeling is the practice of converting real-world problems into mathematical language to make predictions or analyze behavior. In this exercise, the task is to find \( g \) such that \( h = g \circ f \).
This requires abstracting the problem to identify how individual functions combine to represent the given equation, \( h(x) = 2x^2 + x - \sqrt[6]{x} + 1 \). By adjusting and manipulating algebraic forms, we are effectively modeling how the functions interact.
With \( f(x) = \sqrt{x} \), we apply modeling to essentially reverse-engineer \( h \) to discover \( g \). We restate \( h \) as a function of \( y \), then deduce \( g(y) \) to be \( 2y^4 + y^2 - y^{2/6} + 1 \).
Mathematical modeling equips us with tools not only to solve equations but also to visualize relationships between variables and functions to better address and solve practical scenarios. By breaking down a complex real-world function into simpler components, we successfully create models that illuminate the pathways from inputs through intermediate steps to outputs.

• This modeling provides clarity and aids in understanding the dynamic settings formed by function interactions.
• Developing models through clear, step-by-step reasoning is key in applications ranging from engineering to finance.