Given that $$ f(x)=x^{2}-9 x $$ and $$ g(x)=x+12 $$, find (a) $$ f+g=\square $$ and its domain is $$ \square $$ and its domain is $$ \square $$ and its domain is $$ \square $$ (b) $$ f-g=\square $$ and its domain is $$ x
eq \square $$ (c) $$ f g=\square $$ (d) $$ \frac{f}{g}=\square $$

Question

Given that $$ f(x)=x^{2}-9 x $$ and $$ g(x)=x+12 $$, find (a) $$ f+g=\square $$ and its domain is $$ \square $$ and its domain is $$ \square $$ and its domain is $$ \square $$ (b) $$ f-g=\square $$ and its domain is $$ x 
eq \square $$ (c) $$ f g=\square $$ (d) $$ \frac{f}{g}=\square $$

UpStudy AI · Accepted Answer

Let's solve each part step by step.

### Given Functions:
- $$ f(x) = x^{2} - 9x $$
- $$ g(x) = x + 12 $$

---

### (a) $$ f + g $$

**Calculation:**
$$
f(x) + g(x) = (x^{2} - 9x) + (x + 12) = x^{2} - 8x + 12
$$

**Expression:**
$$
f + g = x^{2} - 8x + 12
$$

**Domain:**
Both $$ f(x) $$ and $$ g(x) $$ are polynomials, which are defined for all real numbers.

$$
\text{Domain of } f + g: \mathbb{R}
$$

---

### (b) $$ f - g $$

**Calculation:**
$$
f(x) - g(x) = (x^{2} - 9x) - (x + 12) = x^{2} - 10x - 12
$$

**Expression:**
$$
f - g = x^{2} - 10x - 12
$$

**Domain:**
Both $$ f(x) $$ and $$ g(x) $$ are polynomials, which are defined for all real numbers.

$$
\text{Domain of } f - g: \mathbb{R}
$$

---

### (c) $$ f \cdot g $$

**Calculation:**
$$
f(x) \cdot g(x) = (x^{2} - 9x)(x + 12)
$$
$$
= x^{3} + 12x^{2} - 9x^{2} - 108x
$$
$$
= x^{3} + 3x^{2} - 108x
$$

**Expression:**
$$
f \cdot g = x^{3} + 3x^{2} - 108x
$$

**Domain:**
Both $$ f(x) $$ and $$ g(x) $$ are polynomials, which are defined for all real numbers.

$$
\text{Domain of } f \cdot g: \mathbb{R}
$$

---

### (d) $$ \frac{f}{g} $$

**Calculation:**
$$
\frac{f(x)}{g(x)} = \frac{x^{2} - 9x}{x + 12}
$$

**Expression:**
$$
\frac{f}{g} = \frac{x^{2} - 9x}{x + 12}
$$

**Domain:**
The denominator $$ g(x) = x + 12 $$ must not be zero.

$$
x + 12 \neq 0 \implies x \neq -12
$$

$$
\text{Domain of } \frac{f}{g}: x \in \mathbb{R}, x \neq -12
$$

---

### Summary of Answers:

**(a)**
$$
f + g = x^{2} - 8x + 12
$$
$$
\text{Domain}: \mathbb{R}
$$

**(b)**
$$
f - g = x^{2} - 10x - 12
$$
$$
\text{Domain}: \mathbb{R}
$$

**(c)**
$$
f \cdot g = x^{3} + 3x^{2} - 108x
$$
$$
\text{Domain}: \mathbb{R}
$$

**(d)**
$$
\frac{f}{g} = \frac{x^{2} - 9x}{x + 12}
$$
$$
\text{Domain}: x \neq -12
$$
(a) $$ f + g = x^{2} - 8x + 12 $$, domain: all real numbers (b) $$ f - g = x^{2} - 10x - 12 $$, domain: all real numbers (c) $$ f \cdot g = x^{3} + 3x^{2} - 108x $$, domain: all real numbers (d) $$ \frac{f}{g} = \frac{x^{2} - 9x}{x + 12} $$, domain: $$ x \neq -12 $$

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(b) and its domain is
©
(d)

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