Exercise 5 Let $$ f(x)=1+\frac{1}{x} $$ and $$ g(x)=\frac{1}{x} $$. Find the following: a) $$ (f+g)(x) $$ and its domain. b) $$ (f-g)(x) $$ and its domain. c) $$ (f g)(x) $$ and its domain. d) $$ \left(\frac{f}{g}\right)(x) $$ and its domain. e) $$ (f+g)(3) $$ f) $$ (f g)(2) $$ g) $$ (f \circ g)(x) $$ and its domain. h) $$ (g \circ f)(x) $$ and its domain.

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### a) $$ (f+g)(x) $$ and its domain

The sum of the functions $$ f(x) $$ and $$ g(x) $$ is given by:
$$
(f+g)(x) = f(x) + g(x) = \left(1 + \frac{1}{x}\right) + \frac{1}{x} = 1 + \frac{2}{x}
$$

**Domain:** The domain of $$ f(x) $$ and $$ g(x) $$ is all real numbers except $$ x = 0 $$ (since division by zero is undefined). Therefore, the domain of $$ (f+g)(x) $$ is:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### b) $$ (f-g)(x) $$ and its domain

The difference of the functions $$ f(x) $$ and $$ g(x) $$ is given by:
$$
(f-g)(x) = f(x) - g(x) = \left(1 + \frac{1}{x}\right) - \frac{1}{x} = 1
$$

**Domain:** The domain remains the same as before:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### c) $$ (f g)(x) $$ and its domain

The product of the functions $$ f(x) $$ and $$ g(x) $$ is given by:
$$
(fg)(x) = f(x) \cdot g(x) = \left(1 + \frac{1}{x}\right) \cdot \frac{1}{x} = \frac{1}{x} + \frac{1}{x^2}
$$

**Domain:** The domain is the same:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### d) $$ \left(\frac{f}{g}\right)(x) $$ and its domain

The quotient of the functions $$ f(x) $$ and $$ g(x) $$ is given by:
$$
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{1 + \frac{1}{x}}{\frac{1}{x}} = x + 1
$$

**Domain:** The domain is restricted by $$ g(x) \neq 0 $$, which occurs when $$ x \neq 0 $$:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### e) $$ (f+g)(3) $$

Now we evaluate $$ (f+g)(3) $$:
$$
(f+g)(3) = 1 + \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}
$$

### f) $$ (f g)(2) $$

Now we evaluate $$ (fg)(2) $$:
$$
(fg)(2) = \frac{1}{2} + \frac{1}{2^2} = \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}
$$

### g) $$ (f \circ g)(x) $$ and its domain

The composition $$ (f \circ g)(x) $$ is given by:
$$
(f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right) = 1 + x
$$

**Domain:** The domain is determined by $$ g(x) $$ being defined, which is $$ x \neq 0 $$:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### h) $$ (g \circ f)(x) $$ and its domain

The composition $$ (g \circ f)(x) $$ is given by:
$$
(g \circ f)(x) = g(f(x)) = g\left(1 + \frac{1}{x}\right) = \frac{1}{1 + \frac{1}{x}} = \frac{x}{x + 1}
$$

**Domain:** The domain is determined by $$ f(x) $$ being defined, which is $$ x \neq 0 $$:
$$
\text{Domain: } x \in \mathbb{R}, x \neq 0
$$

### Summary of Results
- a) $$ (f+g)(x) = 1 + \frac{2}{x} $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
- b) $$ (f-g)(x) = 1 $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
- c) $$ (fg)(x) = \frac{1}{x} + \frac{1}{x^2} $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
- d) $$ \left(\frac{f}{g}\right)(x) = x + 1 $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
- e) $$ (f+g)(3) = \frac{5}{3} $$
- f) $$ (fg)(2) = \frac{3}{4} $$
- g) $$ (f \circ g)(x) = 1 + x $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
- h) $$ (g \circ f)(x) = \frac{x}{x + 1} $$, Domain: $$ x \in \mathbb{R}, x \neq 0 $$
a) $$ (f+g)(x) = 1 + \frac{2}{x} $$, Domain: $$ x \neq 0 $$ b) $$ (f-g)(x) = 1 $$, Domain: $$ x \neq 0 $$ c) $$ (fg)(x) = \frac{1}{x} + \frac{1}{x^2} $$, Domain: $$ x \neq 0 $$ d) $$ \left(\frac{f}{g}\right)(x) = x + 1 $$, Domain: $$ x \neq 0 $$ e) $$ (f+g)(3) = \frac{5}{3} $$ f) $$ (fg)(2) = \frac{3}{4} $$ g) $$ (f \circ g)(x) = 1 + x $$, Domain: $$ x \neq 0 $$ h) $$ (g \circ f)(x) = \frac{x}{x + 1} $$, Domain: $$ x \neq 0 $$

Exercise 5 Let and . Find the following:
a) and its domain.
b) and its domain.
c) and its domain.
d) and its domain.
e)
f)
g) and its domain.
h) and its domain.

Solución de inteligencia artificial de Upstudy

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Solución

a) and its domain

b) and its domain

c) and its domain

d) and its domain

e)

f)

g) and its domain

h) and its domain

Summary of Results

Beyond the Answer

a)

b)

c)

d)

e)

f)

g)

h)

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Exercise 5 Let and . Find the following: a) and its domain. b) and its domain. c) and its domain. d) and its domain. e) f) g) and its domain. h) and its domain.