91Ó°ÊÓ

A real root of a polynomial is a value of x that makes the polynomial equal to zero. When a polynomial has a real root, it means there is a real number where the graph of the polynomial intersects the x-axis. One important characteristic of polynomials is that those with an odd degree and real number coefficients are guaranteed to have at least one real root. This occurs because the graph must cross the x-axis due to the nature of odd-degree polynomials.
Let's consider the function in this exercise, \( f(x) = x^5 - 2x^2 + x \). This is a polynomial of degree 5, which is odd. Therefore, according to the theorem for polynomials of odd degrees, it must have at least one real root.

• When \( f(x) \) crosses the x-axis, it indicates a real root.
• This property is useful for proving further characteristics of the function, such as whether it is onto.

An onto function, or a surjective function, is one where every possible output of the function (the range) is achieved by some input from the domain. Here, this means that for every real number \( y \), there should be at least one real number \( x \) such that \( f(x) = y \).
For the function \( f(x) = x^5 - 2x^2 + x \), we know that it is an odd-degree polynomial with real coefficients. This implies the function spans from \( -\infty \) to \( \infty \) because the ends of the graph stretch indefinitely in both directions along the y-axis.

• Thanks to the presence of a real root, we can ensure there is an intersection on the x-axis.
• This adherence to odd-degree patterns guarantees that every value \( y \) can be reached from some value of \( x \).

Thus, the function \( f(x) \) is onto.

To be a one-to-one function, or injective, each output must be the result of exactly one input. In other words, if \( f(x_1) = f(x_2) \), then \( x_1 \) must equal \( x_2 \). To check whether a function is one-to-one, we can use its derivative to assess whether \( f'(x) \) is always positive or negative throughout its domain. This continuous increase or decrease ensures that each \( x \) value corresponds to a unique \( y \) value.
For our polynomial \( f(x) = x^5 - 2x^2 + x \), its derivative is \( f'(x) = 5x^4 - 4x + 1 \).
Notice that \( f'(x) \) is a polynomial with even degree, not maintaining consistent positivity or negativity:

• This derivative changes sign, showing that \( f(x) \) does not always increase or always decrease.
• As a result, multiple \( x \) values may map to the same \( y \) value, making the function not one-to-one.

The conclusion is that function \( f(x) \) is not injective.