The functions f and g are defined by f(x) = 2x2 and g(x) = (3x-2)/(x+1). Find fog(x).

Step 1: Identify the given functions. The functions are defined as: $f(x) = 2x^2$ $g(x) = \frac{3x-2}{x+1}$

Step 2: Understand the composite function $(f \circ g)(x)$. The notation $(f \circ g)(x)$ means $f(g(x))$. This involves substituting the entire function $g(x)$ into $f(x)$ wherever 'x' appears in $f(x)$.

Step 3: Substitute $g(x)$ into $f(x)$. Replace $x$ in $f(x) = 2x^2$ with $g(x) = \frac{3x-2}{x+1}$: $f(g(x)) = 2\left(\frac{3x-2}{x+1}\right)^2$

Step 4: Simplify the expression. Square the fraction: $f(g(x)) = 2 \cdot \frac{(3x-2)^2}{(x+1)^2}$ Expand the numerator $(3x-2)^2$ and the denominator $(x+1)^2$: $(3x-2)^2 = (3x)^2 - 2(3x)(2) + (-2)^2 = 9x^2 - 12x + 4$ $(x+1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1$ Substitute these expanded forms back into the expression: $f(g(x)) = 2 \cdot \frac{9x^2 - 12x + 4}{x^2 + 2x + 1}$ Multiply the numerator by 2: $f(g(x)) = \frac{2(9x^2 - 12x + 4)}{x^2 + 2x + 1}$ $f(g(x)) = \frac{18x^2 - 24x + 8}{x^2 + 2x + 1}$

The composite function is: $\boxed{\frac{18x^2 - 24x + 8}{x^2 + 2x + 1}}$