https://file.aiquickdraw.com/mathbot/file/1729772900670mkxbc97d.jpg

The problem asks to determine whether the following functions are one-to-one, onto, or neither.

### Given functions:
1. \( f: \mathbb{R} \to \mathbb{R}, \, f(x) = |x| \)
2. \( f: \mathbb{R} \to \mathbb{R}, \, f(x) = x^2 \)

Let’s analyze each function.

### 1. \( f(x) = |x| \)

#### One-to-One (Injective):
A function is **one-to-one** if different inputs always result in different outputs, i.e., \( f(x_1) = f(x_2) \implies x_1 = x_2 \).

For \( f(x) = |x| \), let’s check:
- \( f(2) = |2| = 2 \)
- \( f(-2) = |-2| = 2 \)

Since \( f(2) = f(-2) \) but \( 2 \neq -2 \), the function is **not injective**.

#### Onto (Surjective):
A function is **onto** if every element in the codomain (here \( \mathbb{R} \)) is the image of at least one element in the domain (also \( \mathbb{R} \)).

- The function \( f(x) = |x| \) outputs only non-negative real numbers (\( f(x) \geq 0 \) for all \( x \)).
- Therefore, negative numbers in \( \mathbb{R} \) are not in the range.

Thus, \( f(x) = |x| \) is **not surjective** onto \( \mathbb{R} \).

#### Conclusion:
\( f(x) = |x| \) is **neither one-to-one nor onto**.

### 2. \( f(x) = x^2 \)

#### One-to-One (Injective):
For the function \( f(x) = x^2 \), check:
- \( f(2) = 4 \)
- \( f(-2) = 4 \)

Since \( f(2) = f(-2) \) but \( 2 \neq -2 \), the function is **not injective**.

#### Onto (Surjective):
- The function \( f(x) = x^2 \) outputs non-negative real numbers (\( f(x) \geq 0 \)).
- Therefore, negative numbers in \( \mathbb{R} \) are not in the range of \( f(x) \).

Thus, \( f(x) = x^2 \) is **not surjective** onto \( \mathbb{R} \).

#### Conclusion:
\( f(x) = x^2 \) is **neither one-to-one nor onto**.

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. What conditions would make a function both injective and surjective?
2. How can the domain or codomain be adjusted to make the functions above injective or surjective?
3. Can the function \( f(x) = |x| \) be modified to become one-to-one?
4. What is the significance of the range of a function in determining whether it is onto?
5. How can graphical analysis help in determining injectivity and surjectivity?

**Tip:** To check if a function is injective, always try to find distinct inputs that produce the same output. For surjectivity, examine whether every possible output is covered by some input in the domain.

Show whether the following functions are one-to-one, onto, or neither.
1. f: R → R, f(x) = |x|
2. f: R → R, f(x) = x^2

Explore how to determine whether the functions f(x) = |x| and f(x) = x^2 are one-to-one, onto, or neither. This problem involves analyzing injectivity and surjectivity for real numbers and includes key concepts such as absolute value and quadratic functions. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation