Problem 34 - Calculus with Concepts in Calculus

Question

Write \(h\) as the composite \(g \circ f\) of two functions \(f\) and \(g\) (neither of which is equal to \(h\) ). $$ h(x)=\sqrt{[x]-1} $$

Step-by-Step Solution

Answer

\( f(x) = [x] - 1, \ g(u) = \sqrt{u} \) Thus, \( h(x) = g(f(x)) = \sqrt{[x] - 1} \).

Step 1: Identify the Inner Function

Begin by identifying the inner function, which will be the function \( f(x) \). Look for a transformation that is done first in the expression for \( h(x) \). In this context, the expression inside the square root function can be chosen as the inner function. \[ f(x) = [x] - 1 \]

Step 2: Identify the Outer Function

Next, identify the outer function, which will be the function \( g(u) \), where \( u \) is the result of applying \( f(x) \). The remaining part of the original function \( h(x) \) is the square root applied to the output of the inner function. Thus, the outer function \( g \) can be:\[ g(u) = \sqrt{u} \]

Step 3: Define Composite Function

The composite function \( g \circ f \) is defined by applying \( f \) to the input \( x \) and then applying \( g \) to the result of \( f(x) \). Ensure that combining \( f(x) \) and \( g(u) \) gives the original function \( h(x) \):\[ g(f(x)) = g([x] - 1) = \sqrt{[x] - 1} \]This matches the original function \( h(x) \), confirming the correct decomposition.

Key Concepts

Inner Function

Understanding the concept of an inner function is essential in function composition. The inner function is the initial function that you apply to the input value. Think of it as the first domino in a line of dominoes. In the context of the exercise, when we are given a function like \[ h(x) = \sqrt{[x] - 1} \]we need to identify what operation occurs first within the nested operations. Here, the expression inside the square root, \([x] - 1\), is processed first. Therefore, the inner function is \[ f(x) = [x] - 1 \]This function takes the input and performs a specific operation on it, setting up the result for the next layer of processing.

Outer Function

After understanding the inner function, the outer function steps into play. This is the function applied after the inner function has been executed. Essentially, it processes the output of the inner function. If inner functions are the beginning, outer functions are the conclusion of the sequence.In our example, after taking the result from the inner function \( f(x) = [x] - 1 \), we apply the square root to complete the original expression. Hence, the outer function is \[ g(u) = \sqrt{u} \]Here, \( u \) represents the output of the inner function. The outer function finalizes the transformation to yield the composite function that rebuilds the original function \( h(x) \).

Composite Function

Once we have both the inner and outer functions determined, the composite function is formed by applying these in sequence. The composite function, represented as \( g \circ f \), involves inputting \( x \) into \( f(x) \) and then using the result as the input for \( g(u) \).The goal of constructing a composite function is to break down complex operations into more manageable steps, similar to simplifying a complicated task into smaller, actionable tasks.To reconstruct the original function \( h(x) = \sqrt{[x] - 1} \), verify that the output of \( f(x) = [x] - 1 \) when processed through \( g(u) = \sqrt{u} \), results in \[ g(f(x)) = \sqrt{[x] - 1} \]This step confirms that the composite function aligns perfectly with \( h(x) \), ensuring that the decomposition and reassembly of functions were correct.