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Product and Quotient Functions

Given:
[tex]\[ m(x) = x^2 + 4x \][/tex]
[tex]\[ n(x) = x \][/tex]

Find [tex]\((mn)(x)\)[/tex]:

[tex]\[(mn)(x) = (x^2 + 4x) \cdot x\][/tex]
[tex]\[(mn)(x) = x^3 + 4x^2\][/tex]



Answer :

GinnyAnswer
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]

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