Question

name: (last) (first)
multiplication with function notations
perform the indicated operations.
show your work.
circle your final answer
1)
$g(a) = a^2$
$f(a) = 3a + 4$
find $f(a) \cdot g(a)$

2)
$g(a) = 3a + 3$
$f(a) = 2a - 2$
find $(f \cdot g)(a)$

3)
$g(x) = 4x + 3$
$h(x) = x^3 - 2x^2$
find $g(x) \cdot h(x)$

4)
$g(x) = x + 3$
$f(x) = -x + 4$
find $f(x) \cdot g(x)$

Explanation:

Step1: Identify the operation

We need to find the product of the functions \( f(a) \) and \( g(a) \), where \( g(a) = a^2 \) and \( f(a) = 3a + 4 \). The operation is function multiplication, so we multiply the two expressions: \( (3a + 4) \times a^2 \).

Step2: Distribute the \( a^2 \)

Using the distributive property (also known as the distributive law of multiplication over addition), we multiply each term inside the parentheses by \( a^2 \):

$$ (3a + 4) \times a^2 = 3a \times a^2 + 4 \times a^2 $$

Step3: Simplify the terms

For the first term, \( 3a \times a^2 \), we use the rule of exponents \( a^m \times a^n = a^{m + n} \). Here, \( m = 1 \) (since \( a = a^1 \)) and \( n = 2 \), so \( a^1 \times a^2 = a^{1 + 2} = a^3 \). Thus, \( 3a \times a^2 = 3a^3 \).

For the second term, \( 4 \times a^2 = 4a^2 \) (since multiplying a constant by a variable term just attaches the constant to the variable term).

Step4: Combine the simplified terms

Putting the two simplified terms together, we get:

$$ 3a^3 + 4a^2 $$

Answer:

\( 3a^3 + 4a^2 \)