4) f(x)= { 3x^2 x< and equal 1
{ ax+ b x > 1
What is the relation between a and b if the function is continuous for allof x.
To determine the relation between a and b for the given function to be continuous for all x, we need to ensure that the two pieces (3x^2 and ax + b) of the function connect smoothly at x = 1.
For a function to be continuous at x = 1, we need the limit of f(x) as x approaches 1 from the left (denoted as f(1-)) to be equal to the limit of f(x) as x approaches 1 from the right (denoted as f(1+)), and these limits should be equal to the value of f(x) at x = 1 (denoted as f(1)).
Let's calculate these limits:
1. Limit as x approaches 1 from the left (f(1-)):
Taking the limit from the left means we consider the value of the function just before x = 1.
Substituting x = 1 into the first piece of the function:
f(1-) = 3(1)^2 = 3(1) = 3.
2. Limit as x approaches 1 from the right (f(1+)):
Taking the limit from the right means we consider the value of the function just after x = 1.
Substituting x = 1 into the second piece of the function:
f(1+) = a(1) + b = a + b.
3. Value of the function at x = 1 (f(1)):
Substituting x = 1 into the whole function:
f(1) = 3(1)^2 = 3(1) = 3.
To have a continuous function, f(1-) = f(1) = f(1+). Therefore, we equate the values we calculated:
3 = a + b
So, the relation between a and b for the function to be continuous for all x is a + b = 3.