Question 59·Medium·Nonlinear Functions
The function is defined as
If is continuous at , what is the value of ?
For continuity questions with piecewise functions, focus on the boundary point where the formula changes: plug that x-value into each piece, set the resulting expressions equal, and solve for the unknown constant. Pay close attention to which inequality (≤ or >) goes with each piece and be careful with negative signs and basic algebra when solving for the parameter.
Hints
Identify which piece to use at x = -1
Look at the condition under each formula: which one applies at x ≤ -1, and which one at x > -1? You need to use both formulas at x = -1 for continuity.
Think about what continuity means at a point
For a piecewise function to be continuous at a boundary point, the value coming from the left and the value coming from the right at that x must be the same. How can you express that using the two formulas at x = -1?
Compute the left-hand value
Plug x = -1 into the first expression, . Keep track of the negative signs carefully when you square and when you multiply.
Set up and solve an equation for k
After finding the value from the first expression at x = -1, set this equal to the expression evaluated at x = -1. Then solve the resulting equation for k.
Desmos Guide
Use Desmos to solve for k with an equation
In Desmos, type the equation (-1)^2 + 2(-1) = 3(-1) + k. Desmos will treat k as a variable and show the value of k that makes both sides equal; that value is the one that makes the function continuous at x = -1.
Step-by-step Explanation
Use the idea of continuity for a piecewise function
For a piecewise function to be continuous at a point (here, x = -1), the y-values from the left and from the right must match at that x.
So, the value from the first piece at x = -1 must equal the value from the second piece at x = -1.
Find the value from the left side (first piece) at x = -1
For x ≤ -1, the function is given by
Plug in x = -1:
So from the left side, the function approaches -1 at x = -1.
Express the right-hand value in terms of k at x = -1
For x > -1, the function is given by
At x = -1, the expression for the right-hand side would be
Continuity means this right-hand value must equal the left-hand value you just found.
Set the two sides equal and solve for k
Set the left-hand value equal to the right-hand expression:
Now solve for k:
So the value of k that makes f continuous at x = -1 is , which corresponds to answer choice C.