To find the value of the inverse function [tex]\( f^{-1}(-9) \)[/tex] for the given function [tex]\( f(x) = \frac{x+4}{x+6} \)[/tex], follow these steps:
1. Express [tex]\( f(x) \)[/tex] in function notation:
[tex]\[
y = f(x) = \frac{x+4}{x+6}
\][/tex]
2. To find the inverse, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Start with:
[tex]\[
y = \frac{x+4}{x+6}
\][/tex]
Multiply both sides by [tex]\( x + 6 \)[/tex] to get rid of the denominator:
[tex]\[
y(x + 6) = x + 4
\][/tex]
Distribute [tex]\( y \)[/tex]:
[tex]\[
yx + 6y = x + 4
\][/tex]
Isolate the terms involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
yx - x = 4 - 6y
\][/tex]
Factor out [tex]\( x \)[/tex] on the left side:
[tex]\[
x(y - 1) = 4 - 6y
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4 - 6y}{y - 1}
\][/tex]
3. This expression is the inverse function [tex]\( f^{-1}(y) \)[/tex]:
[tex]\[
f^{-1}(y) = \frac{4 - 6y}{y - 1}
\][/tex]
4. To find [tex]\( f^{-1}(-9) \)[/tex], substitute [tex]\( y = -9 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(-9) = \frac{4 - 6(-9)}{-9 - 1}
\][/tex]
Simplify within the numerator and denominator:
[tex]\[
f^{-1}(-9) = \frac{4 + 54}{-10} = \frac{58}{-10} = -5.8
\][/tex]
Therefore, the value of [tex]\( f^{-1}(-9) \)[/tex] is:
[tex]\[
f^{-1}(-9) = -5.8
\][/tex]
