91Ó°ÊÓ

To determine if a function is one-to-one, we need to verify if distinct inputs result in distinct outputs. In other words, a function is one-to-one if, whenever we have two different inputs, we get two different outputs. This can be formally expressed as: if \(f(x_1) = f(x_2)\), then it must be true that \(x_1 = x_2\).

For the given function \(q(x) = \frac{1 - x}{x - 5}\), we need to verify this condition by assuming \(q(x_1) = q(x_2)\) and then proving that \(x_1 = x_2\). If we can show this, we'll know that the function is indeed one-to-one and therefore provides distinct outputs for different inputs.

Once we've assumed that \(q(x_1) = q(x_2)\), the next step is to eliminate the fractions by cross-multiplying. This allows us to simplify the equation and makes it easier to compare the terms.

Starting with \(\frac{1 - x_1}{x_1 - 5} = \frac{1 - x_2}{x_2 - 5}\), we cross-multiply to get: \((1 - x_1)(x_2 - 5) = (1 - x_2)(x_1 - 5)\).

Cross-multiplication helps us clear the denominators and gives us an equation where we can expand and simplify further.

After cross-multiplying, we need to combine and simplify the terms on both sides of the equation to see if we end up with \(x_1 = x_2\). This will verify if the function outputs are distinct for different inputs.

By expanding and simplifying \((1 - x_1)(x_2 - 5) = (1 - x_2)(x_1 - 5)\), we get:
1. Expand both sides: \x_2 - 5 - x_1 x_2 + 5x_1 = x_1 - 5 - x_1 x_2 + 5x_2\
2. Combine like terms: \x_2 - 5 + 5x_1 = x_1 - 5 + 5x_2\
3. Subtract like terms to simplify: \-5 + 5x_1 - 5x_2 = x_1 - x_2\ and \ -5( x_2 - x_1) = -5(x_2 - x_1)\

Since the simplification leads us to conclude that \-5 = -5\, it follows that \(x_1 = x_2\), proving that the function indeed outputs distinct values for distinct inputs. Therefore, the function \(q(x) = \frac{1 - x}{x - 5}\) is one-to-one.