Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The composite function is \(y = (f \circ g)(x)\), where \(g(x) = \sqrt{x+4}\) and \(f(u) = \frac{u-2}{u+2}\).

Step 1: Identify the Inner Function

Notice that the expression under the square roots, i.e., \(x + 4\), is common in both the numerator and the denominator of the given function. We can set this equal to \(u\), so \(u = \sqrt{x+4}\).

Step 2: Rewrite the Expression in Terms of the Inner Function

Substitute \(u\) into the original equation, replacing \(\sqrt{x+4}\) with \(u\). The function becomes: \[ y = \frac{u - 2}{u + 2} \].

Step 3: Define the Composite Functions

Now that the function is expressed in terms of \(u\), identify the two functions that compose the original function. The inner function is \(g(x) = \sqrt{x+4}\), and the outer function is \(f(u) = \frac{u - 2}{u + 2}\).

Step 4: Write the Composite Function

The composite function can be expressed as \(y = (f \circ g)(x) = f(g(x))\). Substitute \(g(x)\) from Step 3 into \(f(u)\), resulting in: \[ y = \frac{\sqrt{x+4} - 2}{\sqrt{x+4} + 2} \].

Key Concepts

Inner Function Identification

In order to solve composite function problems, identifying the inner function is a crucial first step.
By examining the structure of the original expression, we can pinpoint parts of the function that repeat or form a base component.

• Take, for example, the expression under a square root, such as \( x + 4 \) in our exercise.
• This repeating element suggests it can be defined as an inner function.

To make the problem simpler, assign a simple variable to this repeating component. In this case, we assign \( u \) to \( \sqrt{x+4} \).
This approach helps break down complex functions into simpler, manageable components.

Function Substitution

Function substitution involves replacing identified parts of an equation with their equivalent variables.
This is done to rewrite solutions in a simplified form, facilitating further analysis and manipulation.

• In this context, after defining \( u = \sqrt{x+4} \), we substitute \( u \) into the original expression.
• The substitution results in a new equation: \( y = \frac{u - 2}{u + 2} \).

Substitution is a powerful technique because it maintains equality while simplifying the problem, making subsequent steps more straightforward.
By translating the function into terms of \( u \), we import a clearer view of the structure, paving the way for further analysis.

Function Composition

Function composition is an essential mathematical concept involving the combination of two functions.
When looking at composite functions, think of them as operations performed one after another, like a sequence of steps.

• In this exercise, we discern that two functions are composed: an inner function \( g(x) = \sqrt{x+4} \), and an outer function \( f(u) = \frac{u - 2}{u + 2} \).
• The order of application matters where \( g(x) \) is applied first, followed by \( f(u) \).

When expressing composite functions, we use the notation \( (f \circ g)(x) \) which reads as "f composed with g of x".
This indicates that the output of \( g(x) \) becomes the input for \( f(u) \). Function composition simplifies complex equations by modularizing operations.

Algebraic Manipulation

Once the function has been decomposed and substituted, algebraic manipulation plays a critical role in simplifying expressions further.
Manipulating algebraic expressions involves applying operations to achieve a desired structure or output while keeping equations balanced.

• After substitution, we handle the expression \( y = \frac{u - 2}{u + 2} \), derived from our previous steps.
• This involves simplifying fractions, factoring, or performing operations to match initial conditions or solve equations.

Algebraic manipulation helps reveal simplified, equivalent forms of functions that are easier to analyze or use in calculations.
This step is crucial in finding the cleaned-up versions of composite functions necessary in advanced problem-solving.