Questions: For f(x)=x^3-1 and g(x)=4x^2+7, find each of the following: a. f+g b. f-g a. (f+g)(x)=x^3+4x^2+6 (Simplify your answer.) The domain of f+g is (Type your answer in interval notation.)

For f(x)=x^3-1 and g(x)=4x^2+7, find each of the following:
a. f+g
b. f-g
a. (f+g)(x)=x^3+4x^2+6
(Simplify your answer.)
The domain of f+g is
(Type your answer in interval notation.)

Transcript text: For $f(x)=x^{3}-1$ and $g(x)=4 x^{2}+7$, find each of the following: a. $f+g$ b. $f-g$ a. $(f+g)(x)=x^{3}+4 x^{2}+6$ (Simplify your answer.) The domain of $f+g$ is $\square$ (Type your answer in interval notation.)

Solution

Solution Steps

To solve the given problem, we need to perform operations on the functions $ f(x) = x^3 - 1 $ and $ g(x) = 4x^2 + 7 $.

a. To find $ f+g $, we add the expressions for $ f(x) $ and $ g(x) $.

b. To find $ f-g $, we subtract the expression for $ g(x) $ from $ f(x) $.

c. The domain of $ f+g $ is determined by the intersection of the domains of $ f(x) $ and $ g(x) $. Since both functions are polynomials, their domain is all real numbers.

Step 1: Calculate $ f+g $

To find $ f+g $, we add the functions $ f(x) = x^3 - 1 $ and $ g(x) = 4x^2 + 7 $:

\[ f+g = (x^3 - 1) + (4x^2 + 7) = x^3 + 4x^2 + 6 \]

Step 2: Calculate $ f-g $

To find $ f-g $, we subtract $ g(x) $ from $ f(x) $:

\[ f-g = (x^3 - 1) - (4x^2 + 7) = x^3 - 4x^2 - 8 \]

Step 3: Determine the Domain of $ f+g $

The domain of a polynomial function is all real numbers. Since both $ f(x) $ and $ g(x) $ are polynomials, the domain of $ f+g $ is:

\[ (-\infty, \infty) \]