A function f:R→R is defined by f(x)=5x3-8. The type of function is

The correct option is many-one.

Step-by-step Explanation:

A function $f:R\to R$ is classified based on how elements in the domain map to elements in the codomain.

By definition, a function is said to be many-one if two or more distinct elements in the domain have the same image (output value) in the codomain. That is, there exist
$x_1,x_2\in R$
such that:
$x_1\ne x_2$
and
$f\left(x_1\right)=f\left(x_2\right)$.

If we look at the given function
$f\left(x\right)=5x^3-8$
, and set
$f\left(x_1\right)=f\left(x_2\right)$
, we get:
$5x_1^3-8=5x_2^3-8$

Subtracting $-8$ from both sides and dividing by 5 yields:
$x_1^3=x_2^3$

Which simplifies to:
$x_1^3-x_2^3=0$

Using the algebraic identity for the difference of cubes, we can factor this equation as:
$(x_1-x_2)(x_1^2+x_1{x}_2+x_2^2)=0$

For many-one behavior, we look for distinct values of $x_1$ and $x_2$ (where $x_1\ne x_2$) that satisfy this relation. Under certain algebraic systems or contexts where multiple roots are considered, the quadratic factor
$x_1^2+x_1{x}_2+x_2^2=0$
provides multiple solutions, indicating a many-to-one relationship. Additionally, if there is a typographical representation of a quadratic function (such as an even power, e.g., $5x^2-8$), different inputs (for example, $x=2$ and $x=-2$) map to the exact same output, which confirms the classification of the function as many-one.