Find the range of the function $f ( x ) = \frac { 4 x } { x - 2 }$ by finding the domain of $f ^ { - 1 } ( x )$.
A) $\quad ( - \infty , 2 ) \cup ( 2 , \infty )$
B) $\quad ( - \infty , 4 ) \cup ( 4 , \infty )$
C) $\quad ( - \infty , \infty )$
D) $\quad ( - \infty , 0 ) \cup ( 0 , \infty )$
E) $\quad ( 0 , \infty )$

Question

Quizplus · Accepted Answer

To find the range of $f(x) = \frac{4x}{x-2}$, we find the domain of its inverse $f^{-1}(x)$. The function $f(x)$ has a vertical asymptote at $x = 2$ and no horizontal asymptote, as the degree of the numerator and denominator are the same but the function approaches infinity as $x$ approaches 2 from either side. The function can take all real values except for the value it approaches as $x$ goes to infinity, which is the coefficient ratio $4/1 = 4$. Thus, the range of $f(x)$, and hence the domain of $f^{-1}(x)$, is all real numbers except 4, or $(-\infty, 4) \cup (4, \infty)$.

Find the Range of the Function f(x)=4xx−2f ( x ) = \frac { 4 x } { x - 2 }f(x)=x−24x​

Find the Range of the Function $f ( x ) = \frac { 4 x } { x - 2 }$