The linearization of a function $f(x)$ at $a = 1$ is given by the formula for the tangent line:

$L(x) = f(a) + f'(a)(x - a)$

In this case, $f(1) = 2$ and $f'(1) = 2$. So, applying the formula:

$L(x) = f(1) + f'(1)(x - 1) = 2 + 2(x - 1)$

Simplifying:

$L(x) = 2 + 2x - 2 = 2x$

Thus, the linearization of $f$ at $a = 1$ is:

$y = 2x$

So, the correct answer is:

y = 2x

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. How do you find the derivative of a function?
2. What is the difference between linearization and approximation?
3. Can you explain how the linearization works for $a \neq 1$?
4. What is the geometric interpretation of a function’s derivative?
5. How would the linearization change if $f'(1)$ were different?

Tip: Linearization is especially useful for approximating a function near a given point, particularly when the function is too complex to compute directly.