Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \frac { 4 } { 6 x + 5 }$$
A) $f ( x ) : D = \left\{ x \mid x \neq \frac { 2 } { 3 } \right\} , R = \left\{ y \mid y \neq \frac { 4 } { 5 } \right\}$
$$f ^ { - 1 } ( x ) : D = \left\{ x \mid x \neq \frac { 4 } { 5 } \right\} , R = \left\{ y \mid y \neq \frac { 2 } { 3 } \right\}$$
B) $f ( x ) : D = \left\{ x \mid x \neq \frac { 5 } { 6 } \right\} , R = \{ y | y z - 5 \rangle$;
$$f ^ { - 1 } ( x ) : D = \{ x \mid x \neq - 5 \} , R = \left\{ y \mid y \neq \frac { 5 } { 6 } \right\}$$
C) $f ( x )$ : D is all real numbers, $R$ is all real numbers;
$\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ : D is all real numbers, $\mathrm { R }$ is all real numbers

D) $f ( x ) : D = \left\{ x \mid x \neq - \frac { 5 } { 6 } \right\} , R = \{ y \mid y \neq 0 \}$;
$$f ^ { - 1 } ( x ) : D = \{ x \mid x \neq 0 \} , R = \left\{ y \mid y z - \frac { 5 } { 6 } \right\}$$

Question

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \frac { 4 } { 6 x + 5 }$$
A) $f ( x ) : D = \left\{ x \mid x \neq \frac { 2 } { 3 } \right\} , R = \left\{ y \mid y \neq \frac { 4 } { 5 } \right\}$
$$f ^ { - 1 } ( x ) : D = \left\{ x \mid x \neq \frac { 4 } { 5 } \right\} , R = \left\{ y \mid y \neq \frac { 2 } { 3 } \right\}$$
B) $f ( x ) : D = \left\{ x \mid x \neq \frac { 5 } { 6 } \right\} , R = \{ y | y z - 5 \rangle$;
$$f ^ { - 1 } ( x ) : D = \{ x \mid x \neq - 5 \} , R = \left\{ y \mid y \neq \frac { 5 } { 6 } \right\}$$
C) $f ( x )$ : D is all real numbers, $R$ is all real numbers;
$\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ : D is all real numbers, $\mathrm { R }$ is all real numbers

D) $f ( x ) : D = \left\{ x \mid x \neq - \frac { 5 } { 6 } \right\} , R = \{ y \mid y \neq 0 \}$;
$$f ^ { - 1 } ( x ) : D = \{ x \mid x \neq 0 \} , R = \left\{ y \mid y z - \frac { 5 } { 6 } \right\}$$

Quizplus · Accepted Answer

The domain of $f(x) = \frac{4}{6x + 5}$ excludes the value that makes the denominator zero, which is $x = -\frac{5}{6}$. The range of $f(x)$ is all real numbers except 0, because the fraction cannot equal 0. For the inverse function, the domain and range swap roles, so the domain of $f^{-1}(x)$ excludes 0 (since $f(x)$ can never be 0), and the range of $f^{-1}(x)$ excludes $-\frac{5}{6}$, matching the original domain restriction.

Determine I) the Domain of the Function, Ii) the Range f(x)=46x+5f ( x ) = \frac { 4 } { 6 x + 5 }f(x)=6x+54​

Determine I) the Domain of the Function, Ii) the Range $f ( x ) = \frac { 4 } { 6 x + 5 }$