To solve the problem of finding [tex]\((f \cdot g)(x)\)[/tex], where [tex]\(f(x) = 0.5x^2 - 2\)[/tex] and [tex]\(g(x) = 8x^3 + 2\)[/tex], we need to find the product of these two functions.
Let's first rewrite the functions:
[tex]\[ f(x) = 0.5x^2 - 2 \][/tex]
[tex]\[ g(x) = 8x^3 + 2 \][/tex]
The product [tex]\((f \cdot g)(x)\)[/tex] is found by multiplying [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = (0.5x^2 - 2)(8x^3 + 2) \][/tex]
Now, perform the multiplication:
[tex]\[
\begin{align*}
(f \cdot g)(x) &= (0.5x^2 - 2)(8x^3 + 2) \\
&= 0.5x^2 \cdot 8x^3 + 0.5x^2 \cdot 2 - 2 \cdot 8x^3 - 2 \cdot 2 \\
&= 4x^5 + x^2 - 16x^3 - 4 \\
\end{align*}
\][/tex]
Thus, the product function [tex]\((f \cdot g)(x)\)[/tex] can be expressed as:
[tex]\[ (f \cdot g)(x) = 4x^5 - 16x^3 + x^2 - 4 \][/tex]
Therefore, the correct values to fill in are:
[tex]\[ 4.0 \][/tex]
[tex]\[ -16 \][/tex]
[tex]\[ 1.0 \][/tex]
[tex]\[ -4 \][/tex]
So, the complete function is:
[tex]\[ (f \cdot g)(x) = 4.0x^5 - 16x^3 + 1.0x^2 - 4 \][/tex]
Select the correct values for the placeholders from each drop-down menu:
[tex]\[ 4.0 \][/tex]
[tex]\[ -16 \][/tex]
[tex]\[ 1.0 \][/tex]
[tex]\[ -4 \][/tex]
