Find the range of the function $f ( x ) = \frac { 4 x } { x - 5 }$ by finding the domain of $f ^ { - 1 } ( x )$.
A) $\quad ( - \infty , \infty )$
B) $\quad ( - \infty , 0 ) \cup ( 0 , \infty )$
C) $\quad ( - \infty , 4 ) \cup ( 4 , \infty )$
D) $\quad ( 0 , \infty )$
E) $\quad ( - \infty , 5 ) \cup ( 5 , \infty )$

Question

Quizplus · Accepted Answer

To find the range of $f(x) = \frac{4x}{x-5}$, we find the domain of its inverse $f^{-1}(x)$. To do this, we first express $y = \frac{4x}{x-5}$ and then solve for $x$ in terms of $y$. Rearranging, we get $xy - 5y = 4x$, leading to $x(y - 4) = 5y$, and thus $x = \frac{5y}{y-4}$. The domain of this inverse function, which corresponds to the range of the original function, excludes the value that makes the denominator zero, i.e., $y \neq 4$. Therefore, the range of $f(x)$ is $(-\infty, 4) \cup (4, \infty)$.

Find the Range of the Function f(x)=4xx−5f ( x ) = \frac { 4 x } { x - 5 }f(x)=x−54x​

Find the Range of the Function $f ( x ) = \frac { 4 x } { x - 5 }$