To solve this problem, we need to find $g(x)$ given that:

• $\text{fog}(x) = 5x - 2$
• $f(x) = 4x - 3$

Step 1: Express $\text{fog}(x)$ in terms of $f(x)$ and $g(x)$:

The composition $\text{fog}(x)$ means $f(g(x))$. So, we can write:

$f(g(x)) = 5x - 2$

Step 2: Substitute the expression for $f(x)$:

We know $f(x) = 4x - 3$, so we substitute $f(g(x))$ with $4g(x) - 3$:

$4g(x) - 3 = 5x - 2$

Step 3: Solve for $g(x)$:

To isolate $g(x)$, follow these steps:

$4g(x) = 5x - 2 + 3$ $4g(x) = 5x + 1$ $g(x) = \frac{5x + 1}{4}$

Thus, the function $g(x)$ is:

$g(x) = \frac{5x + 1}{4}$

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Would you like more details or clarification on any step? Here are some related questions you may find helpful:

1. How do you handle composition of functions in general?
2. What is the difference between function composition and addition of functions?
3. Can the method above be used for more complex functions?
4. How can you verify your solution for function composition?
5. What is the inverse of the function $f(x) = 4x - 3$?

Tip: Always double-check your composition operations by plugging back into the original equation to verify the result.