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.