To shift the function \( f(x) = 2x^2 - 1 \) to the left by 3 units, you need to replace \( x \) with \( x + 3 \) in the original function. This leads to the new function \( g(x) \).
Here's how it's done:
\[ g(x) = f(x + 3) = 2(x + 3)^2 - 1 \]
Next, we can expand this expression:
\[ g(x) = 2((x + 3)(x + 3)) - 1 \] \[ = 2(x^2 + 6x + 9) - 1 \] \[ = 2x^2 + 12x + 18 - 1 \] \[ = 2x^2 + 12x + 17 \]
Thus, the expression representing \( g(x) \) is:
\[ g(x) = 2x^2 + 12x + 17 \]