We are given the equations:

$x = g(x)$

$g(x) = 5x + 1$

$f(x) = 2x - 3$

Step 1: Solve for $x$ in $x = g(x)$

Since $g(x) = 5x + 1$, we substitute:

$x = 5x + 1$

Rearrange:

$x - 5x = 1$

$-4x = 1$

$x = -\frac{1}{4}$

Step 2: Evaluate $f(x)$ at $x = -\frac{1}{4}$

$f\left(-\frac{1}{4}\right) = 2\left(-\frac{1}{4}\right) - 3$

$= -\frac{2}{4} - 3$

$= -\frac{1}{2} - 3$

$= -\frac{1}{2} - \frac{6}{2}$

$= -\frac{7}{2}$

Final Answer:

$f(x) = -\frac{7}{2} \text{ when } x = -\frac{1}{4}$

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Would you like a more detailed explanation or have any questions?

Here are five related questions to expand on this topic:

1. How do you solve equations involving compositions of functions, such as $f(g(x)) = x$?
2. What is the graphical interpretation of $x = g(x)$?
3. How do fixed points of a function relate to solving $x = g(x)$?
4. How does the inverse function relate to solving $g(x) = x$?
5. What are real-world applications of solving equations like $x = g(x)$?

Tip:

Always check for extraneous solutions when solving equations algebraically, especially when dealing with functions and compositions.