To determine which of the functions shown in the image are one-to-one and onto:

1. One-to-one (Injective):

A function is one-to-one if no two different inputs produce the same output, i.e., it passes the horizontal line test.

• $f_1(x) = x^2$: Not one-to-one because it fails the horizontal line test. For example, $f_1(-1) = f_1(1) = 1$.
• $f_2(x) = 2x$: One-to-one because it passes the horizontal line test (linear function with no repeated $y$-values).
• $f_3(x) = x^3 - 2x^2 - 5x + 6$: Not one-to-one because it fails the horizontal line test (the cubic function has multiple turning points and repeated $y$-values for different $x$).
• $f_4(x) = x^3$: One-to-one because it passes the horizontal line test (a monotonic cubic function).

2. Onto (Surjective):

A function is onto if every possible $y$-value in the range is achieved by some $x$-value in the domain.

• $f_1(x) = x^2$: Not onto for $f: \mathbb{R} \to \mathbb{R}$ because it only produces $y \geq 0$.
• $f_2(x) = 2x$: Onto because it is a linear function with no restrictions, and every real number $y$ is achieved.
• $f_3(x) = x^3 - 2x^2 - 5x + 6$: Onto because cubic polynomials are continuous and have no restrictions on their range (cover all $\mathbb{R}$).
• $f_4(x) = x^3$: Onto for the same reason as $f_3(x)$.

Final Results:

• One-to-one and onto: $f_2(x) = 2x$ and $f_4(x) = x^3$.
• $f_3(x) = x^3 - 2x^2 - 5x + 6$ is onto but not one-to-one.
• $f_1(x) = x^2$ is neither one-to-one nor onto.

Would you like further details on any of these functions? Let me know!

Related Questions:

1. What are the specific intervals where $f_3(x)$ fails the horizontal line test?
2. How can you algebraically verify if a function is onto?
3. What transformations make $f_1(x) = x^2$ one-to-one or onto?
4. How does the derivative help in identifying injectivity for $f_3(x)$?
5. Can we restrict $f_1(x) = x^2$ to a domain to make it one-to-one?

Tip:

When verifying injectivity, always use the horizontal line test or calculate the derivative to check if it is strictly increasing or decreasing over its domain!