To classify the given function [tex]\( f(x) = 2x^2 + 6x - 9 \)[/tex] appropriately, let's analyze the different types of sequences and functions it could be.
1. Geometric Sequence: A geometric sequence has the form [tex]\( a, ar, ar^2, ar^3, \ldots \)[/tex], where each term after the first is found by multiplying the previous term by a constant ratio [tex]\( r \)[/tex]. The function [tex]\( f(x) = 2x^2 + 6x - 9 \)[/tex] does not fit this definition as it involves [tex]\( x \)[/tex] in a polynomial expression rather than a multiplicative pattern.
2. Arithmetic Sequence: An arithmetic sequence has the form [tex]\( a, a + d, a + 2d, a + 3d, \ldots \)[/tex], where each term after the first is found by adding a constant difference [tex]\( d \)[/tex] to the preceding term. Again, [tex]\( f(x) = 2x^2 + 6x - 9 \)[/tex] does not follow this pattern, as it is defined by a quadratic polynomial rather than a linear addition.
3. Function: A function is a relation that maps each element [tex]\( x \)[/tex] from a domain to exactly one element [tex]\( y \)[/tex] in the codomain. In this context, [tex]\( f(x) = 2x^2 + 6x - 9 \)[/tex] is a polynomial function, specifically a quadratic function, which maps any real number [tex]\( x \)[/tex] to a corresponding value [tex]\( y \)[/tex].
Since the given expression does not fit the patterns of a geometric or arithmetic sequence, we can conclude that the correct classification for [tex]\( f(x) = 2x^2 + 6x - 9 \)[/tex] is:
Neither geometric nor arithmetic, but a function.
