Problem 28 - Calculus for Biology and Medicine

Question

(a) Write $$h(x, y)=\sqrt{x+y}$$ as a composition of two functions.

Step-by-Step Solution

Answer

\( h(x, y) = f(g(x,y)) \) with \( f(u) = \sqrt{u} \) and \( g(x,y) = x+y \).

Step 1: Understand the composition requirement

We need to express the function \( h(x, y) = \sqrt{x+y} \) as a composition of two functions, say \( f \) and \( g \), such that \( h(x, y) = (f \circ g)(x,y) = f(g(x,y)) \).

Step 2: Choose an inner function

We select the inner function \( g(x,y) \) to simplify the input to the square root. A good choice is \( g(x,y) = x+y \). This gathers the expressions inside the square root together.

Step 3: Define the outer function

With \( g(x,y) = x+y \), the outer function \( f \) should apply the square root to the input. Hence, define \( f(u) = \sqrt{u} \).

Step 4: Verify the composition

Compose the functions as \( f(g(x,y)) = f(x+y) = \sqrt{x+y} \), which matches the original function \( h(x, y) \). Therefore, the composition is correct.

Key Concepts

Understanding the Inner Function

In the context of function composition, the **inner function** is an essential building block that forms the base of a composite function structure. When given a function like \( h(x, y) = \sqrt{x+y} \), our goal is to express it as a function composition. Here, the inner function will typically handle the initial transformation of input variables.

• The choice of the inner function determines how we preprocess or combine inputs before the final operation.
• For \( h(x, y) \), we selected the inner function \( g(x,y) = x + y \).
• This step involves aggregating or transforming inputs to simplify subsequent operations.

By using \( g(x, y) \) to consolidate \( x \) and \( y \), we reduce the problem to managing a simpler input to the outer function, which will handle the actual square root operation.

Defining the Outer Function

The **outer function** acts as the final step in processing the output from the inner function in a function composition sequence. Put simply, it's the function that directly computes the result after the initial inputs have been transformed by the inner function.

• In our case, after the inner function \( g(x, y) = x + y \) collects and aggregates the terms, the role of the outer function is to perform the square root.
• Defined as \( f(u) = \sqrt{u} \), this outer function takes the sum produced by \( g \) and performs the desired operation — in this instance, square rooting.

The accuracy of this outer function is crucial because it directly impacts the correctness of the final result when performing the composition. The definition of \( f \) completes the pathway from inputs \( x \) and \( y \) to the final output \( h(x, y) \).

Exploring Mathematical Functions

Mathematical functions are fundamental tools in mathematics that map inputs to outputs according to specific rules. They can represent a variety of real-world processes and computations, offering critical insights and simplifying complex operations:

• Functions consist of an input domain (where the input values are drawn from) and a codomain (where the output values land).
• For example, our function \( h(x, y) = \sqrt{x+y} \) takes inputs \( x \) and \( y \) and transforms them into an output that belongs to the set of non-negative real numbers.
• In composition, each function — inner or outer — plays a specific role in processing these inputs, leading to the final computed result.

Understanding these building blocks allows manipulation and combination of functions in multifunction compositions, providing a robust mechanism for modeling complex behaviors and computations in mathematics.