If the original function f(x) = 2x^2-1 is shifted to the left 3 units to make the function g(x), which expression would represent g(x)?

Question

2(x−3)2−1
2 times open paren x minus 3 close paren squared minus 1

2(x+3)2−1
2 times open paren x plus 3 close paren squared minus 1

2x2+2
2 x squared plus 2

2x2−4

Answer 1

To shift the function \( f(x) = 2x^2 - 1 \) to the left by 3 units, you replace \( x \) in the function with \( x + 3 \).

So, the shifted function \( g(x) \) can be written as:

\[ g(x) = f(x + 3) = 2(x + 3)^2 - 1 \]

Now, looking through the provided options:

1. \( 2(x-3)^2 - 1 \): This represents a shift to the right, not the left.
2. \( 2(x+3)^2 - 1 \): This is the correct representation of the leftward shift.
3. \( 2x^2 + 2 \): This is not correct, as it does not represent a shift of the original function.
4. \( 2x^2 - 4 \): This is also not correct for a leftward shift.

Thus, the correct expression for \( g(x) \) is:

\[ \boxed{2(x+3)^2 - 1} \]