For a real valued function [tex]f(x)=2 x+3[/tex], find the values of [tex]f(2.99), f(3.01)[/tex] and [tex]f(3)[/tex]. Is this function continuous at [tex]x =3[/tex]?
Answer (1)
Calculate f ( 2.99 ) by substituting x = 2.99 into f ( x ) = 2 x + 3 , which gives f ( 2.99 ) = 8.98 .
Calculate f ( 3.01 ) by substituting x = 3.01 into f ( x ) = 2 x + 3 , which gives f ( 3.01 ) = 9.02 .
Calculate f ( 3 ) by substituting x = 3 into f ( x ) = 2 x + 3 , which gives f ( 3 ) = 9 .
Verify continuity at x = 3 by checking that lim x → 3 − f ( x ) = lim x → 3 + f ( x ) = f ( 3 ) = 9 , confirming that the function is continuous at x = 3 .
Continuous at x = 3
Explanation
Problem Analysis We are given the function f ( x ) = 2 x + 3 . We need to find the values of f ( 2.99 ) , f ( 3.01 ) , and f ( 3 ) , and then determine if the function is continuous at x = 3 .
Calculate f(2.99) First, let's calculate f ( 2.99 ) by substituting x = 2.99 into the function:
f ( 2.99 ) = 2 ( 2.99 ) + 3 = 5.98 + 3 = 8.98
Calculate f(3.01) Next, let's calculate f ( 3.01 ) by substituting x = 3.01 into the function:
f ( 3.01 ) = 2 ( 3.01 ) + 3 = 6.02 + 3 = 9.02
Calculate f(3) Now, let's calculate f ( 3 ) by substituting x = 3 into the function:
f ( 3 ) = 2 ( 3 ) + 3 = 6 + 3 = 9
Check for Continuity To determine if f ( x ) is continuous at x = 3 , we need to check if the limit of f ( x ) as x approaches 3 exists and is equal to f ( 3 ) . Since f ( x ) is a linear function, it is continuous everywhere. We can verify this by checking the left-hand limit and right-hand limit at x = 3 .
The left-hand limit is:
lim x → 3 − f ( x ) = lim x → 3 − ( 2 x + 3 ) = 2 ( 3 ) + 3 = 9
The right-hand limit is:
lim x → 3 + f ( x ) = lim x → 3 + ( 2 x + 3 ) = 2 ( 3 ) + 3 = 9
Since the left-hand limit, the right-hand limit, and f ( 3 ) are all equal to 9, the function is continuous at x = 3 .
Final Answer Therefore, f ( 2.99 ) = 8.98 , f ( 3.01 ) = 9.02 , f ( 3 ) = 9 , and the function is continuous at x = 3 .
Examples
In manufacturing, a function like f ( x ) = 2 x + 3 could represent the cost of producing x items, where there's a fixed cost of 3 units and a variable cost of 2 units per item. Checking continuity at a specific production level (e.g., x = 3 ) ensures that small changes in production don't lead to sudden, large changes in cost, which is crucial for budgeting and planning. Understanding function values around a point helps in predicting costs for slightly different production quantities, aiding in decision-making.
