For ​f(x)equals=x cubed minus 27x3−27 and ​g(x)equals=2 x squared plus 52x2+5​, find each of the following functions and state its domain.
a. fplus+g
b. fminus−g
c. f times •g
d. StartFraction f Over g EndFraction
f
g
e. StartFraction g Over f EndFraction
g
f

Let's first define the functions more clearly:

\[
f(x) = x^3 - 27
\]
\[
g(x) = 2x^2 + 5
\]

Now, let's calculate each of the requested functions and determine their domains.

### a. \( (f + g)(x) \)

\[
(f + g)(x) = f(x) + g(x) = (x^3 - 27) + (2x^2 + 5)
\]
\[
(f + g)(x) = x^3 + 2x^2 - 27 + 5 = x^3 + 2x^2 - 22
\]

**Domain:** Both \(f(x)\) and \(g(x)\) are polynomial functions, and the domain of a polynomial function is all real numbers (\( \mathbb{R} \)). Therefore, the domain of \( (f + g)(x) \) is:
\[
\boxed{\mathbb{R} \text{ (all real numbers)}}
\]

---

### b. \( (f - g)(x) \)

\[
(f - g)(x) = f(x) - g(x) = (x^3 - 27) - (2x^2 + 5)
\]
\[
(f - g)(x) = x^3 - 2x^2 - 27 - 5 = x^3 - 2x^2 - 32
\]

**Domain:** Since both functions are polynomials, the domain remains all real numbers:
\[
\boxed{\mathbb{R} \text{ (all real numbers)}}
\]

---

### c. \( (f \cdot g)(x) \)

\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x^3 - 27) \cdot (2x^2 + 5)
\]
Let's expand this product:
\[
(f \cdot g)(x) = (x^3)(2x^2 + 5) - 27(2x^2 + 5)
\]
\[
= 2x^5 + 5x^3 - 54x^2 - 135
\]

**Domain:** The domain is still all real numbers since it is a polynomial function:
\[
\boxed{\mathbb{R} \text{ (all real numbers)}}
\]

---

### d. \( \left(\frac{f}{g}\right)(x) \)

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^3 - 27}{2x^2 + 5}
\]

**Domain:** The domain of a rational function excludes values of \(x\) that make the denominator zero. To find those values, solve:
\[
2x^2 + 5 = 0
\]
This equation has no real solutions because \(2x^2 + 5 = 0\) leads to \(x^2 = -\frac{5}{2}\), which is impossible over real numbers. Therefore, the denominator never equals zero.

Thus, the domain is:
\[
\boxed{\mathbb{R} \text{ (all real numbers)}}
\]

---

### e. \( \left(\frac{g}{f}\right)(x) \)

\[
\left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} = \frac{2x^2 + 5}{x^3 - 27}
\]

**Domain:** For this rational function, the denominator must not equal zero. Solve:
\[
x^3 - 27 = 0
\]
\[
x^3 = 27
\]
\[
x = 3
\]

So, the function is undefined at \(x = 3\). Therefore, the domain is:
\[
\boxed{\mathbb{R} \setminus \{3\} \text{ (all real numbers except } x =

For f(x) = x^3 - 27 and g(x) = 2x^2 + 5, find each of the following functions and state its domain.
a. f + g
b. f - g
c. f * g
d. f / g
e. g / f

Learn how to find the sum, difference, product, and quotient of the polynomial functions f(x) = x^3 - 27 and g(x) = 2x^2 + 5. This problem covers key algebraic concepts such as polynomials, rational functions, and their domains. Suitable for high school students in Grades 10-12.

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