Certainly! To find the function [tex]\( g(x) \)[/tex] given [tex]\( g[f(x)] = \frac{9x - 3}{x + 3} \)[/tex] and [tex]\( f(x) = \frac{5x}{x + 3} \)[/tex], let's go through a detailed, step-by-step solution.
1. Express [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = \frac{5x}{x + 3}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y(x + 3) = 5x
\][/tex]
[tex]\[
yx + 3y = 5x
\][/tex]
[tex]\[
3y = 5x - yx
\][/tex]
[tex]\[
3y = x(5 - y)
\][/tex]
[tex]\[
x = \frac{3y}{5 - y}
\][/tex]
3. Substitute [tex]\( x = \frac{3y}{5 - y} \)[/tex] into [tex]\( g[f(x)] \)[/tex]:
[tex]\[
g(y) = g\left(\frac{5x}{x + 3}\right) = \frac{9x - 3}{x + 3}
\][/tex]
With [tex]\( x = \frac{3y}{5 - y} \)[/tex]:
[tex]\[
g(y) = \frac{9 \left(\frac{3y}{5 - y}\right) - 3}{\left(\frac{3y}{5 - y}\right) + 3}
\][/tex]
4. Simplify the expression:
[tex]\[
g(y) = \frac{27y / (5 - y) - 3}{3y / (5 - y) + 3}
\][/tex]
First, simplify the numerator:
[tex]\[
27y / (5 - y) - 3 = \frac{27y - 3(5 - y)}{5 - y} = \frac{27y - 15 + 3y}{5 - y} = \frac{30y - 15}{5 - y}
\][/tex]
Next, simplify the denominator:
[tex]\[
3y / (5 - y) + 3 = \frac{3y + 3(5 - y)}{5 - y} = \frac{3y + 15 - 3y}{5 - y} = \frac{15}{5 - y}
\][/tex]
Now, divide the simplified numerator by the simplified denominator:
[tex]\[
g(y) = \frac{\frac{30y - 15}{5 - y}}{\frac{15}{5 - y}} = \frac{30y - 15}{15} = 2y - 1
\][/tex]
5. Thus, we have:
[tex]\[
g(y) = 2y - 1
\][/tex]
Since [tex]\( y \)[/tex] is just a dummy variable representing [tex]\( x \)[/tex], we can rewrite the final function [tex]\( g(x) \)[/tex] as:
[tex]\[
g(x) = 2x - 1
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = 2x - 1
\][/tex]
