Determine whether the function is one-to-one. $f (x) = \frac{x + 1}{2 x - 3}$

a. Yes, it is one-to-one.
b. No, it isn't one-to-one.

The correct answer is A. To see why, view each of the steps below.

Step 1

Start with the assumption that $LaTeX: f\left(a\right)=f\left(b\right)$ .
$LaTeX: f\left(a\right)=\frac{a+1}{2a-3}=\frac{b+1}{2b-3}=f\left(b\right)$ by definition $LaTeX: g\left(f\left(4\right)\right)$ .

Step 2

Cross multiply
$(a + 1) (2 b - 3) = (b + 1) (2 a - 3)$
$2 a b - 3 a + 2 b - 3 = 2 a b - 3 b + 2 a - 3$

Step 3

Cancel like terms
$- 3 a + 2 b = - 3 b + 2 a$

Step 4

Cross multiply
$(a + 1) (2 b - 3) = (b + 1) (2 a - 3)$
$2 a b - 3 a + 2 b - 3 = 2 a b - 3 b + 2 a - 3$

Step 5

Combine like terms
$- 5 a = - 5 b$

Step 6

So $a = b$
Therefore, $f (x)$ is a one-to-one function.

All Steps

Cross multiply
$(a + 1) (2 b - 3) = (b + 1) (2 a - 3)$
$2 a b - 3 a + 2 b - 3 = 2 a b - 3 b + 2 a - 3$

Cancel like terms
$- 3 a + 2 b = - 3 b + 2 a$

Cross multiply
$(a + 1) (2 b - 3) = (b + 1) (2 a - 3)$
$2 a b - 3 a + 2 b - 3 = 2 a b - 3 b + 2 a - 3$

Combine like terms
$- 5 a = - 5 b$

So $a = b$
Therefore, $f (x)$ is a one-to-one function.