Select the correct answer from each drop-down menu.

If [tex]$f(x)=0.5x^2-2$[/tex] and [tex]$g(x)=8x^3+2$[/tex], find the value of the following function.

[tex](f \cdot g)(x) = 4x^5 - 4x^3 + \square x^2 - 4[/tex]



Answer :

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]