Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt{\frac{2 x-1}{3 x+4}}\)

Step-by-Step Solution

Answer

\(f(x) = \sqrt{x}\) and \(g(x) = \frac{2x-1}{3x+4}\).

Step 1: Identify Outer and Inner Functions

To express the function as a composition, we need to identify the outer function and the inner function. For the function \(h(x) = \sqrt{\frac{2x - 1}{3x + 4}}\), the outer function is a square root function, \(f(u) = \sqrt{u}\).

Step 2: Define the Inner Function

The inner function \(g(x)\) is the expression inside the square root. So we let \(g(x) = \frac{2x - 1}{3x + 4}\). This transforms the exam question function into a composition \(h(x) = f(g(x))\).

Step 3: Express in Composite Function Form

We've determined that \(f(g(x)) = \sqrt{g(x)}\) where \(g(x) = \frac{2x - 1}{3x + 4}\). Thus, \(f(x)\) is \(\sqrt{x}\), and \(g(x)\) is \(\frac{2x - 1}{3x + 4}\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions form the foundation of many mathematical concepts. They consist of numbers, variables, and operations like addition, subtraction, multiplication, and division. Understanding these expressions is crucial when dealing with functions like the one in this exercise.

Let's break down the given expression inside the square root to better understand it: \(\frac{2x - 1}{3x + 4}\). This is a rational algebraic expression, meaning it is the ratio of two polynomials.

The numerator \(2x - 1\) and the denominator \(3x + 4\) each represent a linear polynomial. When working with algebraic expressions in the context of functions, it's essential to know several things:

• **Simplification:** Can the expression be simplified? In our case, the expression is already in its simplest form.
• **Evaluation:** What happens if we substitute a specific value for \(x?\) Evaluate \(\frac{2x - 1}{3x + 4}\) carefully.
• **Domain:** What are the allowable values of \(x?\) Avoid values that make the denominator zero, like \(-\frac{4}{3}\).

Understanding these aspects can aid in effectively manipulating and evaluating expressions that are part of larger mathematical functions.

Breaking Down Composite Functions

Composite functions are made from combining two or more functions into one. They are written as \(f(g(x))\), meaning the output of one function \(g(x)\) becomes the input to another function \(f(x)\).

In the exercise, we identified the composite function \(h(x) = \sqrt{\frac{2x - 1}{3x + 4}}\) as \(h(x) = f(g(x))\). Here, our outer function \(f(u)\) is a square root function, while our inner function \(g(x)\) is a rational expression.

It's essential to understand the step-by-step process of forming composite functions:

• **Identify components:** Break down the function into its constituent parts.
• **Substitution:** Implement substitution by replacing \(g(x)\) into \(f(x)\) as \(f(g(x))\).
• **Evaluate:** Evaluate the composite function for specific \(x\) values, taking note of any restrictions caused by the inner function.

Composite functions allow us to build complex operations from simpler ones, enhancing the ability to model real-world phenomena mathematically.

Exploring Square Root Functions

Square root functions involve the square root operation, often appearing in various mathematical and real-world scenarios. They have the general form \(f(x) = \sqrt{x}\), where \(x\) must be non-negative to have real number outputs.

In this exercise, the square root function \(f(u) = \sqrt{u}\) acts as the outer function in the composite setup. Square root functions have unique properties:

• **Domain and Range:** The domain consists of non-negative values for \(x\). The range consists of non-negative real numbers.
• **Behavior:** \(f(x)\) increases slowly for large \(x\). The curve becomes less steep as \(x\) grows.
• **Graphical representation:** It is half of a parabola lying sideways.

Square root functions are essential in understanding phenomena like quadratic equations or applying scales in various sciences. Being able to integrate them into larger function structures is a key skill in mathematical problem-solving.