To find the product function [tex]\((m \cdot n)(x)\)[/tex], where [tex]\(m(x) = x^2 + 4x\)[/tex] and [tex]\(n(x) = x\)[/tex], we need to calculate:
[tex]\[ (m \cdot n)(x) = m(x) \cdot n(x) \][/tex]
First, let's define the two given functions:
1. [tex]\( m(x) = x^2 + 4x \)[/tex]
2. [tex]\( n(x) = x \)[/tex]
Now, we compute the product function by multiplying these functions together:
[tex]\[ (m \cdot n)(x) = m(x) \cdot n(x) \][/tex]
Substitute [tex]\( m(x) \)[/tex] and [tex]\( n(x) \)[/tex] into the equation:
[tex]\[ (m \cdot n)(x) = (x^2 + 4x) \cdot x \][/tex]
Next, we distribute [tex]\( x \)[/tex] in the expression [tex]\( (x^2 + 4x) \cdot x \)[/tex]:
[tex]\[ (m \cdot n)(x) = x^2 \cdot x + 4x \cdot x \][/tex]
Now, calculate each term:
[tex]\[
x^2 \cdot x = x^3
\][/tex]
[tex]\[
4x \cdot x = 4x^2
\][/tex]
Combine the results:
[tex]\[ (m \cdot n)(x) = x^3 + 4x^2 \][/tex]
So, the product function [tex]\((m \cdot n)(x)\)[/tex] is:
[tex]\[ (m \cdot n)(x) = x^3 + 4x^2 \][/tex]
Thus, we have successfully found the product of the functions [tex]\(m(x)\)[/tex] and [tex]\(n(x)\)[/tex]:
[tex]\[ (m \cdot n)(x) = x^3 + 4x^2 \][/tex]
