Determine whether the given function is one-to-one. If so, find a formula for the inverse.
-$$f ( x ) = \frac { 4 x - 7 } { 2 x + 6 }$$
A) Not a one-to-one function
B) $f ^ { - 1 } ( x ) = \frac { 4 x - 7 } { 2 x + 6 }$
C) $f ^ { - 1 } ( x ) = \frac { - 6 x - 7 } { 2 x - 4 }$
D) $f ^ { - 1 } ( x ) = \frac { 2 x - 4 } { - 6 x - 7 }$

Question

Quizplus · Accepted Answer

To determine whether a function is one-to-one, we must check if different inputs will always produce different outputs. In other words, for any $x_1$ and $x_2$, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

Let's set $f(x_1) = f(x_2)$ and simplify:

\begin{align*}
\frac{4x_1 - 7}{2x_1 + 6} &= \frac{4x_2 - 7}{2x_2 + 6}\
(4x_1 - 7)(2x_2 + 6) &= (4x_2 - 7)(2x_1 + 6)\
8x_1 x_2 + 24x_1 - 14x_2 - 42 &= 8x_1 x_2 + 24x_2 - 14x_1 - 42\
38 &= 0
\end{align*}

This is a contradiction, so there are no values of $x_1$ and $x_2$ that satisfy the equation. Therefore, $f(x)$ is one-to-one.

To find the formula for the inverse, let $y=f(x)$ and switch the roles of $x$ and $y$. Then solve for $y$.

\begin{align*}
y &= \frac{4x-7}{2x+6}\
x &= \frac{4y-7}{2y+6}\
x(2y+6) &= 4y-7\
2xy+6x &= 4y-7\
(2x-4)y &= -6x-7\
y &= \frac{-6x-7}{2x-4}
\end{align*}

Therefore, the formula for the inverse is $f^{-1}(x) = \frac{-6x-7}{2x-4}$. The answer is (C).

Determine Whether the Given Function Is One-To-One f(x)=4x−72x+6f ( x ) = \frac { 4 x - 7 } { 2 x + 6 }f(x)=2x+64x−7​

Determine Whether the Given Function Is One-To-One $f ( x ) = \frac { 4 x - 7 } { 2 x + 6 }$