Finding Value of 'a' for Continuous Function at x = 2
The function \( f(x) \) is defined by three expressions for different ranges of \( x \), and we are asked to find the value of "a" given that the function is continuous at \( x = 2 \).
For a function to be continuous at a point, the left-hand limit (as \( x \) approaches the point from the left), the right-hand limit (as \( x \) approaches the point from the right), and the function's value at the point must all be the same.
Let's evaluate the limit from the left (\( x < 2 \)) and from the right (\( x > 2 \)) as well as the function's value at \( x = 2 \):
- The right-hand limit as \( x \) approaches 2 from the right is given by the expression \( x + a \). As \( x \) approaches 2, this expression becomes \( 2 + a \).
- The function's value at \( x = 2 \) is given as 5.
- The left-hand limit as \( x \) approaches 2 from the left is given by the expression \( 2x + 1 \). As \( x \) approaches 2, this expression becomes \( 2(2) + 1 = 4 + 1 = 5 \).
Since the function is continuous at \( x = 2 \), the right-hand limit must equal the left-hand limit and also the function's value at \( x = 2 \).
So we have the equation \( 2 + a = 5 \).
Solving for \( a \):
\[ a = 5 - 2 \]
\[ a = 3 \]
Therefore, the value of "a" that makes the function continuous at \( x = 2 \) is 3.
