Let's analyze the problem.

The original function is $f(x) = 2x^2 - 1$, and it is shifted 3 units to the left to create a new function $g(x)$.

Key Concept:

When a function $f(x)$ is shifted horizontally:

• Left by $c$: Replace $x$ with $x + c$.
• Right by $c$: Replace $x$ with $x - c$.

Thus, for a shift 3 units to the left, we replace $x$ with $x + 3$ in $f(x)$: $g(x) = 2(x + 3)^2 - 1$

Now, match this with the answer choices:

• A: $2(x + 3)^2 - 1$ (Correct)
• B: $2x^2 - 4$ (Doesn't reflect the shift or match $g(x)$).
• C: $2(x - 3)^2 - 1$ (This is for a shift 3 units to the right, not left).
• D: $2x^2 + 2$ (No shift, and incorrect constant term).

Final Answer:

The correct choice is A: $2(x + 3)^2 - 1$.

Would you like further clarification or a detailed explanation of function transformations?

Related Questions:

1. How do vertical shifts affect the equation of a function?
2. What is the difference between horizontal and vertical transformations of a function?
3. How do we identify whether a function is shifted left or right?
4. How would the function change if it were shifted down by 3 units instead of horizontally?
5. Can transformations like reflections or stretches be combined with shifts?

Tip:

Remember, horizontal transformations always affect the $x$-term, and the direction is opposite to the sign in $(x \pm c)$.