For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >=

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Question: For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?

Options:

Correct Answer: 3

Solution:

Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.

For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?

For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?

Step 1: Identify the two pieces of the function f(x). The first piece is x^2 - 4 for x < c, and the second piece is 3x - 5 for x >= c.
Step 2: To find the value of c that makes the function continuous at x = c, we need to set the two pieces equal to each other at x = c.
Step 3: Write the equation: c^2 - 4 = 3c - 5.
Step 4: Rearrange the equation to one side: c^2 - 3c + 1 = 0.
Step 5: Use the quadratic formula or factor the equation to find the value of c. In this case, solving gives c = 3.
Step 6: Verify that when c = 3, both pieces of the function give the same output, ensuring continuity.

Piecewise Functions – Understanding how to analyze and find continuity in piecewise-defined functions.
Continuity at a Point – Determining the value of a function at a point where the definition changes to ensure continuity.
Solving Equations – Setting two expressions equal to each other and solving for the variable.