Question

What are the domain and range of the function? Please explain.
`f(x)= (2)/(x-3)+4`

A. The domain is x ∈ R, x ≠ 3. The range is f(x) .
B. The domain is x . The range is f(x) .
C. The domain is x ∈ R, x ≠ 3. The range is f(x) ∈ R, f(x) ≠ - 4.

Answer 1

`g(x)=(a)/(x)\\\\\text{The domain:}\\\\x\\eq0\\\\\boxed{D=\{x\ |\ x\in\mathbb{R},\ x\\eq0\}}\\\\\text{The range}:\ \boxed{R=\{y\ |\ y\in\mathbb{R},\ y\\eq0\}}.`

Function Transformations

f(x) + n - translate the graph of f(x) n units up

f(x) - n - translate the graph of f(x) n units down

f(x + n) - translate the graph of f(x) n units left

f(x - n) - translate the graph of f(x) n units right

`g(x)=(2)/(x)\\\\g(x-3)=(2)/(x-3)\\\\g(x-3)+4=(2)/(x-3)+4\\\\f(x)=g(x-3)+4`

translate the graph of
`g(x)=(2)/(x)` 3 units right and 4 units up.

Therefore the domain is:

`\{x\ |\ x\in\mathbb{R},\ x\\eq0+3\}=\{x\ |\ x\in\mathbb{R},\ x\\eq3\}`

and the range is:

`\{y\ |\ y\in\mathbb{R},\ y\\eq0+4\}=\{y\ |\ y\in\mathbb{R},\ y\\eq4\}`

Answer:

A. The domain is x .

The range is f(x) ∈ R, f(x) ≠ 4.