To demonstrate that [tex]\( f(x) = \frac{x-4}{x+2} \)[/tex] can be expressed as [tex]\( f(x) = \frac{-6}{x+2} + 1 \)[/tex], follow these steps:
1. Start with the given function:
[tex]\[
f(x) = \frac{x-4}{x+2}
\][/tex]
2. Rewrite the numerator [tex]\( x - 4 \)[/tex] by adding and subtracting 2:
[tex]\[
f(x) = \frac{x+2 - 6}{x+2}
\][/tex]
3. Split this fraction into two separate fractions:
[tex]\[
f(x) = \frac{x+2}{x+2} - \frac{6}{x+2}
\][/tex]
4. Simplify the first fraction:
[tex]\[
f(x) = 1 - \frac{6}{x+2}
\][/tex]
5. Rearrange the terms to match the desired form:
[tex]\[
f(x) = \frac{-6}{x+2} + 1
\][/tex]
Therefore, we have shown that:
[tex]\[
f(x) = \frac{x-4}{x+2} \implies f(x) = \frac{-6}{x+2} + 1
\][/tex]
The two forms of the function are indeed equivalent.
