To find the inverse of the function [tex]\( g(x) = \frac{7x - 1}{5x + 4} \)[/tex], we follow these steps:
1. Set up the equation for the inverse function:
We start by writing [tex]\( g(x) = y \)[/tex], but we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] since we want to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x = \frac{7y - 1}{5y + 4}
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
Isolate [tex]\( y \)[/tex] by cross-multiplying to get rid of the fraction:
[tex]\[
x(5y + 4) = 7y - 1
\][/tex]
Distribute [tex]\( x \)[/tex]:
[tex]\[
5xy + 4x = 7y - 1
\][/tex]
Move all terms involving [tex]\( y \)[/tex] to one side and constant terms to the other side:
[tex]\[
5xy - 7y = -1 - 4x
\][/tex]
Factor out [tex]\( y \)[/tex]:
[tex]\[
y(5x - 7) = -1 - 4x
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-1 - 4x}{5x - 7}
\][/tex]
So, the inverse function is:
[tex]\[
g^{-1}(x) = \frac{-4x - 1}{5x - 7}
\][/tex]
3. Find the domain of [tex]\( g^{-1}(x) \)[/tex]:
The domain of [tex]\( g^{-1}(x) \)[/tex] consists of all real numbers [tex]\( x \)[/tex] except where the denominator is zero. Solve for [tex]\( x \)[/tex] when [tex]\( 5x - 7 = 0 \)[/tex]:
[tex]\[
5x - 7 = 0 \implies x = \frac{7}{5} = 1.4
\][/tex]
Therefore, the domain of [tex]\( g^{-1}(x) \)[/tex] is all real numbers except [tex]\( x = 1.4 \)[/tex]:
[tex]\[
\text{Domain of } g^{-1}(x): (-\infty, 1.4) \cup (1.4, \infty)
\][/tex]
4. Find the range of [tex]\( g^{-1}(x) \)[/tex]:
The range of [tex]\( g^{-1}(x) \)[/tex] is the domain of [tex]\( g(x) \)[/tex]. To find where the original function [tex]\( g(x) = \frac{7x - 1}{5x + 4} \)[/tex] is undefined, solve for [tex]\( x \)[/tex] when the denominator is zero:
[tex]\[
5x + 4 = 0 \implies x = -\frac{4}{5} = -0.8
\][/tex]
Therefore, the range of [tex]\( g^{-1}(x) \)[/tex] is all real numbers except [tex]\( y = -0.8 \)[/tex]:
[tex]\[
\text{Range of } g^{-1}(x): (-\infty, -0.8) \cup (-0.8, \infty)
\][/tex]
In conclusion:
[tex]\[
g^{-1}(x) = \frac{-4x - 1}{5x - 7}
\][/tex]
[tex]\[
\text{Domain of } g^{-1}(x): (-\infty, 1.4) \cup (1.4, \infty)
\][/tex]
[tex]\[
\text{Range of } g^{-1}(x): (-\infty, -0.8) \cup (-0.8, \infty)
\][/tex]
