Question

What are the domain and range of the function f(x) -

メー4x-12~
x+2
OD:× € R,Ry€ R
D: x € RR: {yeR |y#-2;
OD: X* -4,R: {yeR|y +2;

Answer 1

The function
`\( f(x) = (4x-12)/(x+2) \)` has a domain of all real numbers except -2 and a range of all real numbers except -8, due to division by zero.

The given function is
`\( f(x) = (4x-12)/(x+2) \).` For the domain
`(\(D\))` , any real number
`\(x\)` is acceptable except when the denominator becomes zero. Thus,
`\(D\)` is
`\( \{x \in \mathbb{R} \mid x \\eq -2 \} \),` indicating that the function is undefined at
`\(x = -2\).` On the other hand, the range
`(\(R\))` excludes the value
`\(y = -8\),`as the function is undefined at
`\(x = -4\)` due to a zero denominator, leading to
`\( \{y \in \mathbb{R} \mid y \\eq -8 \} \)` as the range.

To elaborate further, when
`\(x\)` approaches -2 from the left, the function tends toward negative infinity, and when
`\(x\)` approaches -2 from the right, it tends toward positive infinity. The vertical asymptote at
`\(x = -2\)` signifies the restriction in the domain. Additionally, the numerator
`\(4x-12\)` factors as
`\(4(x-3)\),` suggesting a zero at
`\(x = 3\),` which influences the shape of the graph. Overall, the domain excludes -2 to avoid division by zero, and the range avoids -8 due to an undefined value, shaping the function's behavior and defining where it is valid and meaningful in the real number system.