The question asks for the composition of two functions, $g(f(x))$, given the following:

• $f(x) = 3x - 1$
• $g(x) = x^2 - 2x - 5$

We are tasked with finding $g(f(x))$, meaning we substitute $f(x)$ into $g(x)$.

Step-by-step solution:

1. Substitute $f(x) = 3x - 1$ into $g(x)$: $g(f(x)) = g(3x - 1)$

2. Now apply the function $g(x) = x^2 - 2x - 5$, replacing $x$ with $3x - 1$: $g(3x - 1) = (3x - 1)^2 - 2(3x - 1) - 5$

3. Expand the squared term: $(3x - 1)^2 = (3x)^2 - 2(3x)(1) + (1)^2 = 9x^2 - 6x + 1$

4. Simplify the linear term: $-2(3x - 1) = -6x + 2$

5. Now substitute these into the original expression: $g(3x - 1) = 9x^2 - 6x + 1 - 6x + 2 - 5$

6. Simplify the expression: $g(3x - 1) = 9x^2 - 12x + (1 + 2 - 5) = 9x^2 - 12x - 2$

Thus, $g(f(x)) = 9x^2 - 12x - 2$.

Conclusion:

The correct answer is option E: $9x^2 - 12x - 2$.

Do you need more details or have any questions? Here are 5 related questions to expand your understanding:

1. How would you compute $f(g(x))$ for the same functions?
2. Can you describe the general process of function composition?
3. How does expanding binomials help in solving function compositions?
4. What is the difference between $g(f(x))$ and $f(g(x))$?
5. How would you solve this problem if one of the functions were a trigonometric function?

Tip: When composing functions, always substitute one function's result into the other carefully to avoid errors in simplification.