Which of the following functions is **not** a one-to-one function?

Question 6 options:

f: ℝ → ℝ, where f(x) = 6x - 2

f: ℝ → ℝ, where f(x) = 2x³ + 7

f: ℤ → ℤ, where f(n) = 6

f: ℤ → ℤ, where f(n) = n + 7

A **one-to-one function** (also called an injective function) is a function where every distinct input corresponds to a distinct output. In other words, if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

Let's analyze each of the given functions:

1. **\( f(x) = 6x - 2 \)**, where \( f: \mathbb{R} \to \mathbb{R} \):
   - This is a linear function with a non-zero slope (6). Linear functions with non-zero slopes are always one-to-one because different values of \( x \) will give different values of \( f(x) \).
   - **This is a one-to-one function.**

2. **\( f(x) = 2x^3 + 7 \)**, where \( f: \mathbb{R} \to \mathbb{R} \):
   - Cubic functions like \( 2x^3 \) are one-to-one because they are strictly increasing or decreasing (no horizontal segments).
   - **This is a one-to-one function.**

3. **\( f(n) = 6 \)**, where \( f: \mathbb{Z} \to \mathbb{Z} \):
   - This function gives the same output (6) for every input. This means multiple inputs map to the same output.
   - **This is not a one-to-one function.**

4. **\( f(n) = n + 7 \)**, where \( f: \mathbb{Z} \to \mathbb{Z} \):
   - This is a linear function with a slope of 1, so it’s one-to-one. Different inputs will produce different outputs.
   - **This is a one-to-one function.**

### Conclusion:
The function that is **not one-to-one** is \( f(n) = 6 \), where \( f: \mathbb{Z} \to \mathbb{Z} \).

Would you like further details on injective functions or any other topic related to this?

---

Here are 5 related questions for practice:
1. What is a **bijective** function, and how does it differ from an injective function?
2. How can you prove a function is **one-to-one** algebraically?
3. Is the function \( f(x) = x^2 \) one-to-one on \( \mathbb{R} \)?
4. Can a function be one-to-one but not **onto**? Give an example.
5. How does the concept of a **many-to-one** function relate to this topic?

**Tip:** To check if a function is one-to-one, use the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function is not injective.

Which of the following functions is **not** a one-to-one function? Question 6 options: f: ℝ → ℝ, where f(x) = 6x - 2 f: ℝ → ℝ, where f(x) = 2x³ + 7 f: ℤ → ℤ, where f(n) = 6 f: ℤ → ℤ, where f(n) = n + 7

Identify which function is not one-to-one from a set of options including linear, cubic, and constant functions. Learn key concepts such as injective functions and the horizontal line test. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation