Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-60: Find a composite function form for \(y\). $$ y=4+\sqrt{x^{2}+1} $$

Step-by-Step Solution

Answer

The composite function form is \( y = f(g(x)) \) with \( f(u) = 4 + \sqrt{u} \) and \( g(x) = x^2 + 1 \).

Step 1: Define Outer and Inner Functions

To express the function as a composition, we first need to identify or define two functions, one function that can be written in terms of the other. Here, notice that the expression inside the square root and the constant 4 can be separated. Let \( u = x^2 + 1 \). Then, define the outer function as \( f(u) = 4 + \sqrt{u} \).

Step 2: Compose the Functions

Now that we have identified the inner and outer functions, we can express the composite function. Substitute the expression for \( u \) from Step 1 into the outer function. This gives us \( y = f(g(x)) \) where \( g(x) = x^2 + 1 \) and \( f(u) = 4 + \sqrt{u} \).

Step 3: Verify the Composite Function

To ensure correctness, substitute \( g(x) = x^2 + 1 \) into \( f(u) = 4 + \sqrt{u} \) and check that this equals the original function \( y = 4 + \sqrt{x^2 + 1} \). Performing the substitution, we find \( f(g(x)) = 4 + \sqrt{x^2 + 1} \), which matches the given \( y \).

Key Concepts

Outer Function

In the context of composite functions, the outer function is the function that will encompass or "wrap" around another function, known as the inner function. In simple terms, when you perform function composition, the outer function represents the structure into which the inner function's outcomes are directly plugged. In the exercise's example, the outer function is defined as \( f(u) = 4 + \sqrt{u} \). This indicates that once we determine the value of \( u \) using another function, this value is substituted into the outer function.

• The outer function often reflects the final operation that modifies or adjusts the inner results.
• In composite functions, the outside function provides the output frame or context after receiving an input from the inner operations.

Understanding the outer function’s role helps in recognizing how input values are ultimately processed and transformed by the complete composed function. It's like making a sandwich, where the outer function is the bread enclosing the filling provided by the inner function.

Inner Function

The inner function plays a crucial role in function composition by providing the initial values or expressions that are then used by the outer function. In this scenario, the inner function is the formula or expression that operates first when calculating the overall composed function. In the problem you’re working with, the inner function is given as \( g(x) = x^2 + 1 \). This function calculates a new value based on the input \( x \) and provides this result to the outer function. Essentially:

• The inner function establishes the foundational calculation that is needed before any further operations can occur by the outer function.
• Inner functions simplify complex expressions by isolating certain calculations from broader expressions.

Each element of the inner function translates initial input into something the outer function can further interpret, much like preparing the ingredients before they’re added to the rest of a recipe. This isolation helps avoid complications when solving more complex mathematical expressions.

Function Composition

Function composition is a critical concept in mathematics that involves combining two functions to produce a new function. This is achieved by using the output from one function as the input to another. The strategy helps simplify complex functional relationships into manageable steps. In our example, the function composition process takes the results of the inner function \( g(x) = x^2 + 1 \) and plugs them into the outer function \( f(u) = 4 + \sqrt{u} \). The composite function is expressed as \( f(g(x)) \). You can think of it as:

• The concept helps manage complexity by breaking down intricate functions into a sequence of simpler ones.
• Function composition allows for building progressively on calculations, ensuring accuracy and clarity of result.

In essence, composing functions offers a way to streamline calculations, applying stepwise operations to achieve clear, systematic solutions. It's akin to following a multi-part instruction guide, where each step relies on the full completion of the previous one. Mastering function composition enhances both understanding and efficiency in mathematical problem-solving.