Sure! Let's go through the steps to rewrite the function [tex]\( f(x) = \frac{1}{x-2} - 1 \)[/tex] in the form [tex]\( y = \frac{3 - x}{x - 2} \)[/tex].
### Step-by-Step Solution:
1. Define the given function:
[tex]\[
f(x) = \frac{1}{x - 2} - 1
\][/tex]
2. Introduce a new variable [tex]\( y \)[/tex] to represent [tex]\( f(x) \)[/tex]:
[tex]\[
y = \frac{1}{x - 2} - 1
\][/tex]
3. Rewrite the equation to isolate the fraction:
[tex]\[
y + 1 = \frac{1}{x - 2}
\][/tex]
4. To clear the fraction, multiply both sides by [tex]\( x - 2 \)[/tex]:
[tex]\[
(y + 1)(x - 2) = 1
\][/tex]
5. Expand the left side of the equation:
[tex]\[
y(x - 2) + (x - 2) = 1
\][/tex]
6. Distribute the terms on the left side:
[tex]\[
yx - 2y + x - 2 = 1
\][/tex]
7. Combine like terms and simplify:
[tex]\[
yx + x - 2y - 2 = 1
\][/tex]
8. Move all terms involving [tex]\( x \)[/tex] to one side and the constants to the other:
[tex]\[
yx + x = 2y + 3
\][/tex]
9. Factor out [tex]\( x \)[/tex] from the left-hand side:
[tex]\[
x(y + 1) = 2y + 3
\][/tex]
10. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{2y + 3}{y + 1}
\][/tex]
11. Invert the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: Consider this as solving a system where previously [tex]\( x \)[/tex] was dependent on [tex]\( y \)[/tex], now [tex]\( y \)[/tex] is dependent on [tex]\( x \)[/tex].
12. Rewrite for clarity:
[tex]\[
y = \frac{3 - x}{x - 2}
\][/tex]
Hence, the transformation shows that the given function [tex]\( f(x) = \frac{1}{x - 2} - 1 \)[/tex] can be rewritten as [tex]\( y = \frac{3 - x}{x - 2} \)[/tex] as required.
