Problem 33 - Precalculus

Question

Write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.) $$h(x)=\sqrt{2 x-1}$$

Step-by-Step Solution

Answer

\(h(x) = f(g(x))\) where \(f(u) = \sqrt{u}\) and \(g(x) = 2x - 1\).

Step 1: Identify the Outer Function

First, look at the expression \(h(x)=\sqrt{2x-1}\). The outermost operation is the square root. Let's denote the outer function as \(f(u) = \sqrt{u}\). This means that whatever is inside the square root will be our inner function.

Step 2: Define the Inner Function

To find the inner function, observe what is inside the square root in the function \(h(x)\). The inner expression is \(2x-1\). Let's denote this inner function as \(g(x) = 2x-1\).

Step 3: Express h(x) as a Composition

Using the functions defined in the previous steps, you can express \(h(x)\) as a composition of \(f(u)\) and \(g(x)\). Specifically, you have:\[ h(x) = f(g(x)) = f(2x-1) = \sqrt{2x-1} \]

Step 4: Verify the Composition

Verify the composition by substituting \(g(x)\) into \(f(u)\):- Compute \(g(x) = 2x - 1\).- Substitute into \(f(u)\): \(f(g(x)) = f(2x-1) = \sqrt{2x-1}\).This matches \(h(x) = \sqrt{2x-1}\), confirming the correctness of our composition.

Key Concepts

Outer Function

When dealing with function composition, it's crucial to identify the **outer function** first. The outer function is the one that operates after the inner function has done its work. In other words, once the inner function processes the input, the result is then passed onto the outer function for further computation. For the given function, \( h(x) = \sqrt{2x-1} \), the outermost operation involves taking a square root. Thus, we denote the outer function as \( f(u) = \sqrt{u} \). Here, \( u \) represents the input that our outer function receives, which in this context is the outcome of the inner function. What makes the outer function stand out in any function composition is its position as the final operation. It's much like the icing on a cake — the last step in the process that turns a completed product into a finished masterpiece.

Inner Function

The **inner function** plays a pivotal role in function composition. It processes the independent variable or input before passing the result to the outer function.
In our exercise, the inner function resides inside the square root, which is \( 2x-1 \). Let's call this function \( g(x) = 2x-1 \). Here, the \( 2x-1 \) is the transformation that takes place first. It operates directly on \( x \), modifying it before the outer function, which is the square root, can act upon the new value.
Understanding the inner function is crucial because it helps set the stage for the outer function’s action. It ensures that the right transformation is applied before further operations commence. By isolating and understanding each function, it becomes easier to manage and work with complex compositions in general.

Square Root Function

The **square root function** is a type of radical function denoted by \( \sqrt{} \). It is a fundamental operation in mathematics represented by the radical symbol (\( \sqrt{} \)). In our given function \( h(x) = \sqrt{2x-1} \), it acts as the outer function. This is because, in the order of operations, the square root is applied after \( 2x-1 \) has been processed by the inner function \( g(x) \).Solving square root functions often involves understanding their properties. Some key points include:

• The square root function only produces non-negative outputs when dealing with real numbers since negative numbers do not have real square roots.
• Functions under a square root must be non-negative. This affects the domain (possible inputs) of the function since it requires the expression under the root to always be greater than or equal to zero.
• When dealing with compositions like \( h(x) = \sqrt{2x-1} \), ensuring that the input from the inner function yields valid results is crucial, particularly to avoid undefined mathematical operations.

Understanding these aspects helps in analyzing, simplifying, and correctly interpreting the behavior and domain of functions with square roots.