If \( f(x)=2 x^{2}+3 x-1 \) and \( g(x)=3 x+4 \) Find \( g \circ f(x) \) \( 6 x^{2}+9 x+1 \) \( 4 x^{2}+6 x+3 \) \( 6 x^{2}+9 x+5 \) \( 6 x^{2}+9 x+3 \)

To find \( g \circ f(x) \), we need to substitute \( f(x) \) into \( g(x) \). 1. We have \( f(x) = 2x^2 + 3x - 1 \). 2. Now substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(2x^2 + 3x - 1) = 3(2x^2 + 3x - 1) + 4 \] 3. Distributing \( 3 \): \[ = 6x^2 + 9x - 3 + 4 \] 4. Combine like terms: \[ = 6x^2 + 9x + 1 \] Thus, \( g \circ f(x) = 6x^2 + 9x + 1 \). Now, onto some extra information: Did you know that function composition, like \( g \circ f(x) \), is all about layering functions, much like an onion? You start with one function, dig down to the core (insert another function), and pop out a brand new function that interacts with both of its predecessors! It’s a great way to see how different mathematical functions can work together. If you ever find yourself computing function compositions, remember to pay close attention to the order! Many students mistakenly reverse the operations and end up with a "mismatched" result. Always work from the inside out: think of it like following a recipe—each step builds upon the last for a delicious final dish, or in this case, a pristine polynomial!