What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
Practice Questions
1 question
Q1
What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
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Setting 2(3) + a = 5 gives a = -1.
Questions & Step-by-step Solutions
1 item
Q
Q: What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
Solution: Setting 2(3) + a = 5 gives a = -1.
Steps: 7
Step 1: Identify the function f(x) which has different expressions based on the value of x.
Step 2: Recognize that for the function to be continuous at x = 3, the value of f(x) as x approaches 3 from the left (2x + a) must equal the value of f(x) at x = 3 (which is 5).
Step 3: Write the left-hand limit as x approaches 3: f(3) = 2(3) + a.
Step 4: Calculate 2(3) which equals 6, so we have 6 + a.
Step 5: Set the left-hand limit equal to the value of the function at x = 3: 6 + a = 5.
Step 6: Solve for a by subtracting 6 from both sides: a = 5 - 6.
