Describe the following cell-surface modifications using the table below.
Cell-surface modification Type(s) of epithelia with the modification Function (include a specific organ) Cilia Goblet cells Microvilli
Use the given input/output table to evaluate or solve the following
X f(x) 3 10 2 4 4 0 7 8 NOTE: In each answer blank space Blackboard is expecting a numerical value. f(4) = f|(4) =
The Correct Answer and Explanation is:
Let’s break this question into two parts as requested:
Part 1: Cell-Surface Modifications Table
| Cell-surface modification | Type(s) of epithelia with the modification | Function (include a specific organ) |
|---|---|---|
| Cilia | Pseudostratified columnar epithelium | Moves mucus and trapped particles; e.g., in the trachea, cilia sweep mucus toward the throat to be expelled. |
| Goblet cells | Simple columnar and pseudostratified columnar epithelium | Secrete mucus to lubricate and protect; e.g., in the respiratory tract, mucus traps dust and microbes. |
| Microvilli | Simple columnar epithelium | Increases surface area for absorption; e.g., in the small intestine, microvilli enhance nutrient absorption. |
Part 2: Function Table Evaluation
Given:
| X | f(x) |
|---|---|
| 3 | 10 |
| 2 | 4 |
| 4 | 0 |
| 7 | 8 |
We are to find:
- f(4)
- f⁻¹(4) (read as “f inverse of 4”)
✅ Answer:
- f(4) = 0
- f⁻¹(4) = 2
🔍 Explanation (300+ words):
To solve problems involving a function table, it’s essential to understand the basic concept of what a function and its inverse represent.
A function, denoted f(x), maps an input x to a single output f(x). In our case, we are given a finite set of such mappings in a table. To find f(4), we simply locate x = 4 in the table and identify the value paired with it. According to the data:
- When x = 4, the function value is f(4) = 0.
This tells us that if we input 4 into the function f(x), the result is 0.
The inverse of a function, written as f⁻¹(y), essentially reverses the operation. Instead of asking “what is the output for a given input?”, it asks “what input gives this output?”
So, f⁻¹(4) asks: for which value of x is f(x) = 4?
Looking at the table, we find that:
- When x = 2, f(2) = 4.
Thus, f⁻¹(4) = 2, because f(2) yields the output 4.
This back-and-forth relationship is the essence of inverse functions: if f(x) = y, then f⁻¹(y) = x.
Summary:
- f(4) means look at the output when the input is 4.
- f⁻¹(4) means look at the input that gives 4 as the output.
Using the table, we find:
- f(4) = 0
- f⁻¹(4) = 2
Understanding how to read and interpret function tables is a fundamental skill in algebra and helps build a strong foundation for working with more complex functions, graphs, and equations.