Problem 44 - Algebra and Trigonometry

Question

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=\sqrt{x}, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[3]{x} $$

Step-by-Step Solution

Answer

\( f \circ g \circ h = \sqrt{\frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1}} \).

Step 1: Understand Composition

Recall that the composite function \( f \circ g \circ h \) means \( f(g(h(x))) \). We need to apply these functions in the order from inside out—starting with \( h(x) \), then applying \( g(x) \), and finally \( f(x) \).

Step 2: Apply Function h

First, apply \( h(x) = \sqrt[3]{x} \). The composition becomes \( f(g(h(x))) = f(g(\sqrt[3]{x})) \).

Step 3: Apply Function g

Next, substitute \( \sqrt[3]{x} \) into \( g(x) = \frac{x}{x-1} \):\[g(\sqrt[3]{x}) = \frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1}.\] The composition now is \( f \left( \frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1} \right) \).

Step 4: Apply Function f

Finally, apply the function \( f(x) = \sqrt{x} \) to the result of \( g \):\[f \left( \frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1} \right) = \sqrt{\frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1}}.\] This expression represents the composite function \( f \circ g \circ h \).

Key Concepts

Function Composition

Function composition is a fundamental concept in calculus and algebra that involves joining two or more functions to form a new function. Think of it as stacking operations on a value, one after another.

When you encounter an expression like \( f \circ g \circ h \), it means you need to apply these functions in sequence, starting from the innermost and moving outward. In simpler terms, evaluate \( h(x) \) first, then take that result and plug it into \( g(x) \), and finally use the outcome in \( f(x) \).

Understanding function composition is crucial because it helps in simplifying complex expressions and solving equations that involve multiple functions. It's also essential for understanding more advanced topics like limits, derivatives, and integrals.

Radical Functions

Radical functions include any function that contains a radical symbol, usually a square root or cube root, for instance. These functions dictate the relationship between the input and output through the concept of root extraction.

In the original exercise, the functions \( f(x) = \sqrt{x} \) and \( h(x) = \sqrt[3]{x} \) include square and cube roots, respectively. Here’s what they mean:

• \( \sqrt{x} \) computes the non-negative square root, providing a function that grows as \( x \) increases.
• \( \sqrt[3]{x} \) calculates the cube root, which can handle both positive and negative numbers due to its symmetric property around the origin.

Grasping how these functions operate is important because they regularly appear in scientific and engineering problems, as well as any situation involving power transformations.

Function Operations

Function operations extend beyond composition. They involve several processes that allow us to manipulate and use functions in various ways, making function operations a diverse and powerful tool.

Apart from composition, typical function operations include addition, subtraction, multiplication, and division of functions. For example, you can sum two functions as \( (f+g)(x) = f(x) + g(x) \) or multiply them as \( (f \cdot g)(x) = f(x) \cdot g(x) \).

In composition, as seen in the problem, function operations allow complex expressions like \( \frac{\sqrt[3]{x}}{\sqrt[3]{x} - 1} \), achieved through substituting and simplifying functions step by step. Understanding how to navigate between these operations is key in simplifying and solving intricate mathematical problems.