Question

Please help:State the implied domain and range: f(x)= 1/x−4 +3

Answer 1

`Domain:(-\infty,\text{ 4\rparen U\lparen4, }\infty),\text{ \textbraceleft x \mid x}4\rbrace`
`Range:\text{ \lparen-}\infty,\text{ 3\rparen U\lparen3, }\infty),\text{ \textbraceleft y \mid y}3\rbrace`

Step-by-step explanation

Given:

`f(x)\text{ = }\frac{1}{x\text{ - 4}}+\text{ 3}`

Desired Outcome:

1. Domain

2. Range

Determine the domain of the function

Note: We want to include only input values that do not require the denominator to be 0 when there is a denominator. As a result, we'll set the numerator to 0 and solve for x.

`\begin{gathered} 0\text{ = x - 4} \\ x\text{ = 4} \end{gathered}`

We'll now remove 4 from the domain. All of the answers are real numbers x > 4 or x < 4. We will also use Union notation to combine the two sets.

That is:

`Domain:(-\infty,\text{ 4\rparen U\lparen4, }\infty),\text{ \textbraceleft x \mid x}4\rbrace`

Determine the range of the function

Set x to 4 to find y

`\begin{gathered} y\text{ = }(1)/(4-4)\text{ + 3} \\ y\text{ = 3} \end{gathered}`

We'll also remove 3 from the range. All of the answers are real numbers y > 3 or y < 3. We will also use Union notation to combine the two sets.

That is:

`Range:\text{ \lparen-}\infty,\text{ 3\rparen U\lparen3, }\infty),\text{ \textbraceleft y \mid y}3\rbrace`