To determine the degree of the composition of two functions, [tex]\( (f \circ g)(x) \)[/tex], we need to follow a series of steps:
1. Identify the degrees of each function:
- The function [tex]\( f(x) = 3x^2 \)[/tex] is a quadratic function, where the highest power of [tex]\( x \)[/tex] is 2. Thus, the degree of [tex]\( f(x) \)[/tex] is 2.
- The function [tex]\( g(x) = 4x^3 + 1 \)[/tex] is a cubic function, where the highest power of [tex]\( x \)[/tex] is 3. Thus, the degree of [tex]\( g(x) \)[/tex] is 3.
2. Understand the composition of functions:
- The composition of [tex]\( f \)[/tex] and [tex]\( g \)[/tex], denoted as [tex]\( (f \circ g)(x) \)[/tex], means that you substitute the function [tex]\( g(x) \)[/tex] into the function [tex]\( f(x) \)[/tex].
3. Determine the degree of the composite function:
- When you substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex], the resulting function will be [tex]\( f(g(x)) \)[/tex].
- Since [tex]\( g(x) \)[/tex] has a degree of 3, we substitute this into [tex]\( f(x) \)[/tex], which has a degree of 2. Thus, the overall degree of the composition [tex]\( (f \circ g)(x) \)[/tex] is given by the product of the degrees of [tex]\( f \)[/tex] and [tex]\( g \)[/tex].
4. Calculate the degree:
- The degree of [tex]\( f \circ g \)[/tex] is [tex]\( 2 \times 3 = 6 \)[/tex].
Therefore, the degree of [tex]\( (f \circ g)(x) \)[/tex] is [tex]\( 6 \)[/tex]. Hence, the correct answer is:
6
