Find all the constants a that make the function fÝxÞ = 3Ýa?xÞ if x < 3 a 2 ?x 2 if x ³ 3 continuous for all x. (2 Points)

To find the constants a that make the function f(x) = 3(a-x) if x < 3 and a^2 - x^2 if x ≥ 3 continuous for all x, we need to ensure that the two parts of the function meet at x = 3. This means that the value of the function at x = 3 for the first part should be equal to the value of the function at x = 3 for the second part.

Step 1: Set the two parts of the function equal to each other at x = 3.

3(a - 3) = a^2 - 3^2

Step 2: Simplify the equation.

3a - 9 = a^2 - 9

Step 3: Rearrange the equation to solve for a.

a^2 - 3a = 0

Step 4: Factor the equation.

a(a - 3) = 0

Step 5: Set each factor equal to zero and solve for a.

a = 0 or a = 3

So, the constants a that make the function continuous for all x are 0 and 3.