https://file.aiquickdraw.com/mathbot/file/1726077221087mx1ojlpq.jpg

Let's analyze each function to determine if they are one-to-one:

### Definition:
A function is **one-to-one (injective)** if every element of the domain maps to a unique element in the codomain. In simpler terms, no two different inputs give the same output.

#### (A) \( f(x) = \sqrt{x - 1} \)
- The square root function is not one-to-one because it only gives non-negative outputs, and values like \( f(2) = f(3) = 1 \), meaning different inputs can give the same output.
- **Not one-to-one.**

#### (B) \( f(x) = 5.6^x \)
- Exponential functions like \( a^x \) (where \( a > 1 \)) are always one-to-one because they are strictly increasing, meaning each input \( x_1 \neq x_2 \) will give distinct outputs.
- **One-to-one.**

#### (C) \( f(x) = \log_3{x} \)
- Logarithmic functions are the inverse of exponential functions, and they are also strictly increasing for \( x > 0 \), meaning different inputs give different outputs.
- **One-to-one.**

#### (D) \( f(x) = 3x - 3 \)
- This is a linear function with a non-zero slope. Linear functions with non-zero slopes are one-to-one because they are strictly increasing or decreasing.
- **One-to-one.**

#### (E) \( f(x) = 3x^2 + 5 \)
- The function is quadratic, which is not one-to-one. A parabola opens upward or downward, and different inputs can yield the same output (e.g., \( f(-1) = f(1) \)).
- **Not one-to-one.**

### Final Answer:
The one-to-one functions are **B, C, and D**.

Would you like further details or have any questions?

---

Here are 5 related questions you might explore:
1. How do we determine if a quadratic function is one-to-one?
2. Can all exponential functions be classified as one-to-one?
3. How do the properties of logarithmic functions relate to one-to-one criteria?
4. Why does the square root function fail to be one-to-one?
5. What role does the derivative play in determining whether a function is one-to-one?

**Tip:** A quick way to test if a function is one-to-one is using the **horizontal line test**: if any horizontal line crosses the graph more than once, the function is not one-to-one.

Which of the following are one-to-one functions? Select all that apply.

This page explains how to determine if functions such as f(x) = √(x - 1), f(x) = 5.6^x, f(x) = log_3(x), and more are one-to-one using mathematical concepts like the horizontal line test and injective criteria. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation