Question

name
5.5 notetaking with vocabulary (continued)
extra practice
in exercises 1–4, find ((f + g)(x)) and ((f - g)(x)) and state the domain of each.
then evaluate (f + g) and (f - g) for the given value of (x).

1. (f(x)=-\frac{1}{2}sqrt3{x}, g(x)=\frac{9}{2}sqrt3{x}; x = - 1000)
2. (f(x)=-x^{2}-3x + 8, g(x)=6x-3x^{2}; x=-1)
3. (f(x)=4x^{3}+12, g(x)=2x^{2}-3x^{3}+9; x = 2)
4. (f(x)=5sqrt4{x}+1, g(x)=-3sqrt4{x}-2; x = 1)

Explanation:

Let's solve problem 3: \( f(x) = 4x^3 + 12 \), \( g(x) = 2x^2 - 3x^3 + 9 \); \( x = 2 \)

Step 1: Find \( (f + g)(x) \)

To find \( (f + g)(x) \), we add the two functions \( f(x) \) and \( g(x) \).

\[

$$\begin{align*} (f + g)(x) &= f(x) + g(x)\\ &= (4x^3 + 12) + (2x^2 - 3x^3 + 9)\\ &= 4x^3 - 3x^3 + 2x^2 + 12 + 9\\ &= x^3 + 2x^2 + 21 \end{align*}$$

\]

Step 2: Find \( (f - g)(x) \)

To find \( (f - g)(x) \), we subtract \( g(x) \) from \( f(x) \).

\[

$$\begin{align*} (f - g)(x) &= f(x) - g(x)\\ &= (4x^3 + 12) - (2x^2 - 3x^3 + 9)\\ &= 4x^3 + 12 - 2x^2 + 3x^3 - 9\\ &= 4x^3 + 3x^3 - 2x^2 + 12 - 9\\ &= 7x^3 - 2x^2 + 3 \end{align*}$$

\]

Step 3: Determine the domain of \( (f + g)(x) \) and \( (f - g)(x) \)

Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial function is all real numbers, \( (-\infty, \infty) \). So the domain of \( (f + g)(x) \) and \( (f - g)(x) \) is also all real numbers.

Step 4: Evaluate \( (f + g)(2) \)

Substitute \( x = 2 \) into \( (f + g)(x) = x^3 + 2x^2 + 21 \):

\[

$$\begin{align*} (f + g)(2) &= (2)^3 + 2(2)^2 + 21\\ &= 8 + 2(4) + 21\\ &= 8 + 8 + 21\\ &= 37 \end{align*}$$

\]

Step 5: Evaluate \( (f - g)(2) \)

Substitute \( x = 2 \) into \( (f - g)(x) = 7x^3 - 2x^2 + 3 \):

\[

$$\begin{align*} (f - g)(2) &= 7(2)^3 - 2(2)^2 + 3\\ &= 7(8) - 2(4) + 3\\ &= 56 - 8 + 3\\ &= 51 \end{align*}$$

\]

Final Answers:

• \( (f + g)(x) = x^3 + 2x^2 + 21 \), Domain: \( (-\infty, \infty) \), \( (f + g)(2) = 37 \)
• \( (f - g)(x) = 7x^3 - 2x^2 + 3 \), Domain: \( (-\infty, \infty) \), \( (f - g)(2) = 51 \)

Answer:

Let's solve problem 3: \( f(x) = 4x^3 + 12 \), \( g(x) = 2x^2 - 3x^3 + 9 \); \( x = 2 \)

Step 1: Find \( (f + g)(x) \)

To find \( (f + g)(x) \), we add the two functions \( f(x) \) and \( g(x) \).

\[

$$\begin{align*} (f + g)(x) &= f(x) + g(x)\\ &= (4x^3 + 12) + (2x^2 - 3x^3 + 9)\\ &= 4x^3 - 3x^3 + 2x^2 + 12 + 9\\ &= x^3 + 2x^2 + 21 \end{align*}$$

\]

Step 2: Find \( (f - g)(x) \)

To find \( (f - g)(x) \), we subtract \( g(x) \) from \( f(x) \).

\[

$$\begin{align*} (f - g)(x) &= f(x) - g(x)\\ &= (4x^3 + 12) - (2x^2 - 3x^3 + 9)\\ &= 4x^3 + 12 - 2x^2 + 3x^3 - 9\\ &= 4x^3 + 3x^3 - 2x^2 + 12 - 9\\ &= 7x^3 - 2x^2 + 3 \end{align*}$$

\]

Step 3: Determine the domain of \( (f + g)(x) \) and \( (f - g)(x) \)

Both \( f(x) \) and \( g(x) \) are polynomials. The domain of a polynomial function is all real numbers, \( (-\infty, \infty) \). So the domain of \( (f + g)(x) \) and \( (f - g)(x) \) is also all real numbers.

Step 4: Evaluate \( (f + g)(2) \)

Substitute \( x = 2 \) into \( (f + g)(x) = x^3 + 2x^2 + 21 \):

\[

$$\begin{align*} (f + g)(2) &= (2)^3 + 2(2)^2 + 21\\ &= 8 + 2(4) + 21\\ &= 8 + 8 + 21\\ &= 37 \end{align*}$$

\]

Step 5: Evaluate \( (f - g)(2) \)

Substitute \( x = 2 \) into \( (f - g)(x) = 7x^3 - 2x^2 + 3 \):

\[

$$\begin{align*} (f - g)(2) &= 7(2)^3 - 2(2)^2 + 3\\ &= 7(8) - 2(4) + 3\\ &= 56 - 8 + 3\\ &= 51 \end{align*}$$

\]

Final Answers:

• \( (f + g)(x) = x^3 + 2x^2 + 21 \), Domain: \( (-\infty, \infty) \), \( (f + g)(2) = 37 \)
• \( (f - g)(x) = 7x^3 - 2x^2 + 3 \), Domain: \( (-\infty, \infty) \), \( (f - g)(2) = 51 \)