The functions [tex]f[/tex] and [tex]g[/tex] are defined as [tex]f(x)=x^3[/tex] and [tex]g(x)=4x^2+21x-18[/tex].

a) Find the domain of [tex]f[/tex], [tex]g[/tex], [tex]f+g[/tex], [tex]f-g[/tex], [tex]fg[/tex], [tex]ff[/tex], [tex]\frac{f}{g}[/tex], and [tex]\frac{g}{f}[/tex].

b) Find [tex](f+g)(x)[/tex], [tex](f-g)(x)[/tex], [tex](fg)(x)[/tex], [tex](ff)(x)[/tex], [tex]\left(\frac{f}{g}\right)(x)[/tex], and [tex]\left(\frac{g}{f}\right)(x)[/tex].

a) The domain of [tex]f[/tex] is [tex]\square[/tex].
(Type your answer in interval notation.)



Answer :

Let's tackle part (a) first:

a) Finding the domain of \( f, g, f+g, f-g, fg, ff, \frac{f}{g}, \frac{g}{f} \).

To begin with the basic definitions of the functions:
- \( f(x) = x^3 \)
- \( g(x) = 4x^2 + 21x - 18 \)

#### Domain Definitions:
- The domain of a polynomial function is all real numbers (\( \mathbb{R} \)).

Thus:
- The domain of \( f(x) \) = \( \mathbb{R} \) because it is a polynomial function.
- The domain of \( g(x) \) = \( \mathbb{R} \) because it is a polynomial function.

Now, let's find the domain of the combined functions:

1. \( f+g \):
[tex]\[ (f+g)(x) = f(x) + g(x) = x^3 + 4x^2 + 21x - 18 \][/tex]
The domain of \( f+g \) is \( \mathbb{R} \) since both \( f \) and \( g \) have the domain \( \mathbb{R} \).

2. \( f-g \):
[tex]\[ (f-g)(x) = f(x) - g(x) = x^3 - (4x^2 + 21x - 18) = x^3 - 4x^2 - 21x + 18 \][/tex]
The domain of \( f-g \) is \( \mathbb{R} \).

3. \( fg \):
[tex]\[ (fg)(x) = f(x) \cdot g(x) = x^3 \cdot (4x^2 + 21x - 18) \][/tex]
The domain of \( fg \) is \( \mathbb{R} \).

4. \( ff \):
[tex]\[ (ff)(x) = f(f(x)) = f(x^3) = (x^3)^3 = x^9 \][/tex]
The domain of \( ff \) is \( \mathbb{R} \).

5. \( \frac{f}{g} \):
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{x^3}{4x^2 + 21x - 18} \][/tex]
We need to exclude values of \( x \) for which the denominator is zero, i.e., the solutions to \( 4x^2 + 21x - 18 = 0 \).

To find these zeros, we solve the quadratic equation \( 4x^2 + 21x - 18 = 0 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
[tex]\[ a = 4, \, b = 21, \, c = -18 \][/tex]
[tex]\[ x = \frac{-21 \pm \sqrt{21^2 - 4 \cdot 4 \cdot (-18)}}{2 \cdot 4} = \frac{-21 \pm \sqrt{441 + 288}}{8} = \frac{-21 \pm \sqrt{729}}{8} = \frac{-21 \pm 27}{8} \][/tex]
[tex]\[ x = \frac{6}{8} = \frac{3}{4} \quad \text{or} \quad x = \frac{-48}{8} = -6 \][/tex]

Therefore, the domain of \( \frac{f}{g} \) is \( \mathbb{R} \setminus \left\{ -6, \frac{3}{4} \right\} \).

6. \( \frac{g}{f} \):
[tex]\[ \left( \frac{g}{f} \right)(x) = \frac{4x^2 + 21x - 18}{x^3} \][/tex]
We need to exclude values of \( x \) where the denominator is zero, i.e., \( x = 0 \).

Thus, the domain of \( \frac{g}{f} \) is \( \mathbb{R} \setminus \{ 0 \} \).

To summarize:
- The domain of \( f \) is \( \mathbb{R} \).
- The domain of \( g \) is \( \mathbb{R} \).
- The domain of \( f+g \) is \( \mathbb{R} \).
- The domain of \( f-g \) is \( \mathbb{R} \).
- The domain of \( fg \) is \( \mathbb{R} \).
- The domain of \( ff \) is \( \mathbb{R} \).
- The domain of \( \frac{f}{g} \) is \( \mathbb{R} \setminus \left\{ -6, \frac{3}{4} \right\} \).
- The domain of \( \frac{g}{f} \) is \( \mathbb{R} \setminus \{ 0 \} \).

b) Finding the expressions for \( (f+g)(x), (f-g)(x), (fg)(x), (ff)(x), \left( \frac{f}{g} \right)(x), \left( \frac{g}{f} \right)(x) \):

1. \( (f+g)(x) \):
[tex]\[ (f+g)(x) = f(x) + g(x) = x^3 + 4x^2 + 21x - 18 \][/tex]

2. \( (f-g)(x) \):
[tex]\[ (f-g)(x) = f(x) - g(x) = x^3 - (4x^2 + 21x - 18) = x^3 - 4x^2 - 21x + 18 \][/tex]

3. \( (fg)(x) \):
[tex]\[ (fg)(x) = f(x) \cdot g(x) = x^3 \cdot (4x^2 + 21x - 18) = 4x^5 + 21x^4 - 18x^3 \][/tex]

4. \( (ff)(x) \):
[tex]\[ (ff)(x) = f(f(x)) = f(x^3) = (x^3)^3 = x^9 \][/tex]

5. \( \left( \frac{f}{g} \right)(x) \):
[tex]\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x^3}{4x^2 + 21x - 18} \][/tex]

6. \( \left( \frac{g}{f} \right)(x) \):
[tex]\[ \left( \frac{g}{f} \right)(x) = \frac{g(x)}{f(x)} = \frac{4x^2 + 21x - 18}{x^3} \][/tex]

So, here are the final expressions and their domains accordingly:
- \( (f+g)(x) = x^3 + 4x^2 + 21x - 18 \) with domain \( \mathbb{R} \).
- \( (f-g)(x) = x^3 - 4x^2 - 21x + 18 \) with domain \( \mathbb{R} \).
- \( (fg)(x) = 4x^5 + 21x^4 - 18x^3 \) with domain \( \mathbb{R} \).
- \( (ff)(x) = x^9 \) with domain \( \mathbb{R} \).
- \( \left( \frac{f}{g} \right)(x) = \frac{x^3}{4x^2 + 21x - 18} \) with domain \( \mathbb{R} \setminus \left\{ -6, \frac{3}{4} \right\} \).
- [tex]\( \left( \frac{g}{f} \right)(x) = \frac{4x^2 + 21x - 18}{x^3} \)[/tex] with domain [tex]\( \mathbb{R} \setminus \{ 0 \} \)[/tex].