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)
Question
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)
Solution
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.
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