Question
\( f(a)=a+2 \) \( g(a)=-3 a^{3}-4+a \) Find \( (f \cdot g)(a) \)
Ask by Gardner Graham. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( (f \cdot g)(a) = -3a^4 - 6a^3 + a^2 - 2a - 8 \)
Solution
To find \( (f \cdot g)(a) \), we need to multiply the functions \( f(a) \) and \( g(a) \) together.
Given:
\( f(a) = a + 2 \)
\( g(a) = -3a^3 - 4 + a \)
To find \( (f \cdot g)(a) \), we multiply \( f(a) \) and \( g(a) \) together:
\( (f \cdot g)(a) = f(a) \cdot g(a) \)
Substitute the expressions for \( f(a) \) and \( g(a) \):
\( (f \cdot g)(a) = (a + 2) \cdot (-3a^3 - 4 + a) \)
Now, we can simplify this expression by multiplying the terms together.
Simplify the expression by following steps:
- step0: Solution:
\(\left(a+2\right)\left(-3a^{3}-4+a\right)\)
- step1: Apply the distributive property:
\(a\left(-3a^{3}\right)-a\times 4+a\times a+2\left(-3a^{3}\right)-2\times 4+2a\)
- step2: Multiply the terms:
\(-3a^{4}-4a+a^{2}-6a^{3}-8+2a\)
- step3: Add the terms:
\(-3a^{4}-2a+a^{2}-6a^{3}-8\)
The simplified expression for \( (f \cdot g)(a) \) is:
\( (f \cdot g)(a) = -3a^4 - 2a + a^2 - 6a^3 - 8 \)
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Simplify this solution Mind Expander
To find \( (f \cdot g)(a) \), you need to multiply the functions \( f(a) \) and \( g(a) \). First, we have: \[ f(a) = a + 2 \] \[ g(a) = -3a^3 - 4 + a \] Now, multiply them together: \[ (f \cdot g)(a) = f(a) \cdot g(a) = (a + 2)(-3a^3 - 4 + a) \] Now, let's distribute \( f(a) \) over \( g(a) \): \[ = (a + 2)(-3a^3) + (a + 2)(-4) + (a + 2)(a) \] \[ = -3a^4 - 6a^3 - 4a - 8 + a^2 + 2a \] Combine like terms: \[ = -3a^4 - 6a^3 + (a^2 - 4a + 2a) - 8 \] \[ = -3a^4 - 6a^3 + a^2 - 2a - 8 \] Thus, the product \( (f \cdot g)(a) \) is: \[ \boxed{-3a^4 - 6a^3 + a^2 - 2a - 8} \]
