Problem 35 - The Calculus with Analytic Geometry

Question

Find formulas for \((f \circ g)(x)\) if $$ f(x)=\left\\{\begin{array}{ll} 0 & \text { if } x<0 \\ 2 x & \text { if } 0 \leq x \\ 0 & \text { if } x>1 \end{array} \quad \text { and } \quad g(x)= \begin{cases}1 & \text { if } x<0 \\\ \frac{1}{2} x & \text { if } 0 \leq x \leq 1 \\ 1 & \text { if } x>1\end{cases}\right. $$

Step-by-Step Solution

Answer

(f ∘ g)(x) = 0 if x < 0 or x > 1, and (f ∘ g)(x) = x if 0 ≤ x ≤ 1.

Step 1: Understand the individual functions

First, we need to understand the defined functions. For any given input x, the function f(x) behaves differently based on the value range of x. Similarly, g(x) operates differently for three ranges of x values.

Step 2: Determine the output of g(x)

Determine the output of the function g(x) for different ranges of x:- If x < 0, then g(x) = 1- If 0 ≤ x ≤ 1, then g(x) = \( \frac{1}{2} x \)- If x > 1, then g(x) = 1

Step 3: Substitute g(x) into f(x)

Now substitute g(x) into f(x) to find (f ∘ g)(x):- For x < 0: g(x) = 1 Then f(g(x)) = f(1) = 0 (since 1 > 1) So, (f ∘ g)(x) = 0- For 0 ≤ x ≤ 1: g(x) = \( \frac{1}{2} x \) Then f(g(x)) = f(\( \frac{1}{2} x \)) Since \( 0 \leq \frac{1}{2} x \leq 1 \), f(\( \frac{1}{2} x \)) = 2(\( \frac{1}{2} x \)) = x So, (f ∘ g)(x) = x- For x > 1: g(x) = 1 Then f(g(x)) = f(1) = 0 (since 1 > 1) So, (f ∘ g)(x) = 0

Step 4: Combine the results

Combine the results from different ranges of x values:- If x < 0, then (f ∘ g)(x) = 0- If 0 ≤ x ≤ 1, then (f ∘ g)(x) = x- If x > 1, then (f ∘ g)(x) = 0

Key Concepts

Piecewise Functions

A piecewise function is a function defined by multiple sub-functions. Each sub-function is valid for a particular interval of the domain. For example, the function \(f(x)\) mentioned is defined differently based on the value of \(x\):

• If \(x < 0\), \(f(x) = 0\)
• If \(0 \leq x \leq 1\), \(f(x) = 2x\)
• If \(x > 1\), \(f(x) = 0\)

Piecewise functions enable us to model situations where a rule or relationship changes depending on the interval. These could represent real-world scenarios like tax brackets, regulations, or physical phenomena.

Composition of Functions

The composition of functions involves applying one function to the results of another function. This is denoted as \((f \circ g)(x)\), which means \(f(g(x))\). The process involves taking an input \(x\), calculating \(g(x)\), and then using that output as the input for \(f\).
For the given functions:
\[ f(x) = \begin{cases} 0 & \text{if } x < 0 \ 2x & \text{if } 0 \leq x \leq 1 \ 0 & \text{if } x > 1 \end{cases} \ g(x) = \begin{cases} 1 & \text{if } x < 0 \ \frac{1}{2} x & \text{if } 0 \leq x \leq 1 \ 1 & \text{if } x > 1 \end{cases} \] We find \((f \circ g)(x)\) by substituting \(g(x)\) into \(f(x)\).

Evaluating Composite Functions

To evaluate composite functions, we follow these steps:

• Step 1: Understand the Individual Functions: Understand how each function operates across different intervals.
• Step 2: Determine the Output of \(g(x)\): Calculate the output of \(g(x)\) for each interval.
• Step 3: Substitute \(g(x)\) into \(f(x)\): Use the output from \(g(x)\) as input for \(f(x)\).
• Step 4: Combine the Results: Gather the final results for each interval.

Applying these steps helps ensure a correct and comprehensive solution. For the problem at hand, the composite function is:

• If \(x < 0\), \(g(x) = 1\), and \(f(g(x)) = 0\) because the input is greater than 1. So, \((f \circ g)(x) = 0\).
• If \(0 \leq x \leq 1\), \(g(x) = \frac{1}{2} x\), and \(f(g(x)) = x\). So, \((f \circ g)(x) = x\).
• If \(x > 1\), \(g(x) = 1\), and \(f(g(x)) = 0\) because the input is greater than 1. So, \((f \circ g)(x) = 0\).

Combining these results gives us the final composite function:

• \((f \circ g)(x) = 0\) for \(x < 0\)
• \((f \circ g)(x) = x\) for \(0 \leq x \leq 1\)
• \((f \circ g)(x) = 0\) for \(x > 1\)