To solve the problem in the image, let's analyze the information provided:

We have the following functions:

• $f(x) = 4x - 3$
• $g(x) = \frac{8x - 7}{8x - 5}$
• $(f \circ g)(x) = 4x - 1$

The composition $(f \circ g)(x)$ means that $f(g(x))$ should equal $4x - 1$.

Step 1: Substitute $g(x)$ into $f(x)$.

$f(g(x)) = 4\left(\frac{8x - 7}{8x - 5}\right) - 3$

Step 2: Simplify the expression.

Let's distribute the 4 into the fraction: $f(g(x)) = \frac{4(8x - 7)}{8x - 5} - 3 = \frac{32x - 28}{8x - 5} - 3$

Step 3: Combine the expression to match $4x - 1$.

$f(g(x)) = \frac{32x - 28 - 3(8x - 5)}{8x - 5}$

Simplifying the numerator: $f(g(x)) = \frac{32x - 28 - 24x + 15}{8x - 5}$ $f(g(x)) = \frac{8x - 13}{8x - 5}$

For this to equal $4x - 1$, the expression $\frac{8x - 13}{8x - 5}$ must equal $4x - 1$.

This doesn't match directly with the expression $4x - 1$. However, from the problem context, $g(x)$ was directly provided and the composition gives us another expression for $g(x)$.

Conclusion:

There appears to be a contradiction or a mismatch in the function composition provided. It could be that additional context or clarification is required to match these conditions exactly.

Would you like to explore this further or have any specific questions on this step?

Related Questions:

1. How would the equation $f(g(x)) = 4x - 1$ change if $f(x)$ were a different linear function?
2. What happens to the function composition if $g(x)$ is a more complex function, like a quadratic?
3. How does the domain of $g(x)$ affect the overall composition function $(f \circ g)(x)$?
4. Can $g(x)$ be simplified to match $4x - 1$ under specific conditions?
5. What would $g(x)$ look like if we wanted $(f \circ g)(x)$ to be a constant function?

Tip: When solving function compositions, always check the domain of the composed function to ensure it's valid across all values of $x$.