The problem asks us to find $f(x-1)$ given the function $f(x) = 3x^4 - 5x^2 + 9$.

1. Problem Description

The problem asks us to find $f(x-1)$ given the function $f(x) = 3x^4 - 5x^2 + 9$.

2. Solution Steps

To find $f(x-1)$, we substitute $x-1$ for $x$ in the expression for $f(x)$.

$f(x-1) = 3(x-1)^4 - 5(x-1)^2 + 9$

We can expand the terms:

$(x-1)^2 = x^2 - 2x + 1$

$(x-1)^4 = ((x-1)^2)^2 = (x^2 - 2x + 1)^2 = (x^2 - 2x + 1)(x^2 - 2x + 1) = x^4 - 2x^3 + x^2 - 2x^3 + 4x^2 - 2x + x^2 - 2x + 1 = x^4 - 4x^3 + 6x^2 - 4x + 1$

So,

$f(x-1) = 3(x^4 - 4x^3 + 6x^2 - 4x + 1) - 5(x^2 - 2x + 1) + 9$

$f(x-1) = 3x^4 - 12x^3 + 18x^2 - 12x + 3 - 5x^2 + 10x - 5 + 9$

$f(x-1) = 3x^4 - 12x^3 + (18-5)x^2 + (-12+10)x + (3-5+9)$

$f(x-1) = 3x^4 - 12x^3 + 13x^2 - 2x + 7$

3. Final Answer

$f(x-1) = 3x^4 - 12x^3 + 13x^2 - 2x + 7$