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The image shows a math assignment titled **Math 1LS3 Assignment 2** under "Section 1.4 (geese) 0.3 (elephants): Working with functions." The questions are as follows:

1. (a) Define the composition \( m \circ p \) of functions \( m \) and \( p \).
   
   (b) Look at Example 1.4.2 (geese) 0.3.2 (elephants). What can you say about the product of an increasing function and a decreasing function?
   
   (c) What is the purpose of the horizontal line test? State the test.
   
   (d) Define the term: inverse function.
   
   (e) Write down the cancellation formulas for a function and its inverse.

### Solutions:

1. **(a) Definition of composition \( m \circ p \):**
   The composition of two functions \( m \) and \( p \), denoted \( m \circ p \), is defined as applying function \( p \) first and then applying function \( m \) to the result. Formally:
   \[
   (m \circ p)(x) = m(p(x))
   \]
   
2. **(b) Product of an increasing function and a decreasing function:**
   When an increasing function is multiplied by a decreasing function, the product does not follow a strict pattern (e.g., increasing or decreasing) and depends on the behavior of the specific functions involved. However, typically, the product can exhibit both increasing and decreasing behavior over different intervals.

3. **(c) Horizontal line test:**
   The purpose of the horizontal line test is to determine whether a function is *injective* (one-to-one). The test states: If no horizontal line intersects the graph of a function more than once, then the function is injective and has an inverse.

4. **(d) Definition of inverse function:**
   An inverse function \( f^{-1} \) of a function \( f \) is a function that reverses the operation of \( f \). That is, for every \( y = f(x) \), the inverse function satisfies \( x = f^{-1}(y) \). Formally:
   \[
   f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x
   \]
   for all \( x \) in the domain of \( f^{-1} \) and the range of \( f \).

5. **(e) Cancellation formulas for a function and its inverse:**
   The cancellation formulas are:
   \[
   f(f^{-1}(x)) = x \quad \text{for all } x \text{ in the range of } f
   \]
   \[
   f^{-1}(f(x)) = x \quad \text{for all } x \text{ in the domain of } f
   \]

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Would you like further clarification on any of these answers or examples?

### Related Questions:
1. Can a composition of two injective functions be non-injective? Why or why not?
2. How does the vertical line test differ from the horizontal line test?
3. Provide an example of a function and its inverse.
4. Why must a function be injective for an inverse to exist?
5. What happens to the composition of two decreasing functions?

### Tip:
When dealing with compositions of functions, it is important to pay attention to the domains and ranges of both functions, as this affects whether the composition is well-defined.

Define the composition m o p of functions m and p. What can you say about the product of an increasing function and a decreasing function? What is the purpose of the horizontal line test? Define the term: inverse function. Write down the cancellation formulas for a function and its inverse.

This assignment focuses on key topics in function analysis, including the composition of functions, the horizontal line test for injectivity, and inverse functions. Learn how to apply these concepts through real examples, with detailed explanations on the cancellation formulas for a function and its inverse.

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