Question Statement Given the piecewise function:
f ( x ) = ⎩ ⎨ ⎧ m x − n 5 2 m x + n , x < 1 , x = 1 , x > 1
Determine the values of m n f ( x ) x = 1
Background and Explanation For a function to be continuous at a point, the left-hand limit, right-hand limit, and the function value at that point must all be equal. This problem requires calculating the one-sided limits of the piecewise function and solving the resulting system of equations.
Solution For continuity at x = 1
lim x → 1 − f ( x ) = lim x → 1 + f ( x ) = f ( 1 )
Left-hand limit (using f ( x ) = m x − n x < 1 lim x → 1 − f ( x ) = m ( 1 ) − n = m − n
Right-hand limit (using f ( x ) = 2 m x + n x > 1 lim x → 1 + f ( x ) = 2 m ( 1 ) + n = 2 m + n
Function value at x = 1
f ( 1 ) = 5
Equating the limits to the function value:
m − n = 5 f ( 1 ) 2 m + n = 5 f ( 1 )
Add equations (1) and (2) to eliminate n ( m − n ) + ( 2 m + n ) = 5 + 5 3 m = 10 m = 3 10
Substitute m = 3 10 n
3 10 − 5 3 10 − 15 3 − 5 = n = n = n
Therefore, the values are:
m = 3 10 and n = − 3 5
Continuity condition at a point : lim x → a − f ( x ) = lim x → a + f ( x ) = f ( a ) Left-hand limit calculation : lim x → 1 − ( m x − n ) = m − n Right-hand limit calculation : lim x → 1 + ( 2 m x + n ) = 2 m + n Solving simultaneous equations : Elimination method to solve for two unknowns
Summary of Steps
Identify the left-hand limit as x → 1 m − n
Identify the right-hand limit as x → 1 2 m + n
Set both limits equal to f ( 1 ) = 5
Solve the system by adding equations to eliminate n m = 3 10
Substitute m = 3 10 m − n = 5 n = 3 10 − 5 = − 3 5