Question

find the inverse of the function.
\\( h(x) = \frac{3x + 3}{-5x - 3} \\)
write your answer in the form \\( \frac{ax + b}{cx + d} \\), where \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are integers.
\\( h^{-1}(x) = \boxed{\frac{\boxed{0}}{\boxed{0}}} \\)

Explanation:

Step1: Replace \( h(x) \) with \( y \)

We start by writing the function as \( y = \frac{3x + 3}{-5x - 3} \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap \( x \) and \( y \), so we get \( x = \frac{3y + 3}{-5y - 3} \).

Step3: Solve for \( y \)

First, multiply both sides by \( -5y - 3 \) to eliminate the denominator:
\( x(-5y - 3) = 3y + 3 \)
Expand the left side:
\( -5xy - 3x = 3y + 3 \)
Now, collect terms with \( y \) on one side and the rest on the other side. Add \( 5xy \) to both sides and subtract 3 from both sides:
\( -3x - 3 = 3y + 5xy \)
Factor out \( y \) on the right side:
\( -3x - 3 = y(3 + 5x) \)
Now, divide both sides by \( 3 + 5x \) to solve for \( y \):
\( y = \frac{-3x - 3}{5x + 3} \)
We can factor out a -3 from the numerator:
\( y = \frac{-3(x + 1)}{5x + 3} \) or we can write it as \( y=\frac{ - 3x-3}{5x + 3}\)

Answer:

\( \frac{-3x - 3}{5x + 3} \) (or equivalent forms like \( \frac{-3(x + 1)}{5x + 3} \))