Complete the following steps for each function.


c. State the interval(s) of continuity.


f(x)={2x if x<1
x^2+3x if x≥1; a=1

Verified step by step guidance

1. Step 1: Understand the piecewise function. The function f(x) is defined as two separate expressions depending on the value of x. For x < 1, f(x) = 2x. For x \(\geq\) 1, f(x) = x^2 + 3x.
2. Step 2: Determine the continuity at the point where the function changes, which is at x = 1. 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.
3. Step 3: Calculate the left-hand limit as x approaches 1 from the left (x \(\to\) 1^-). Use the expression for x < 1, which is 2x. Evaluate the limit: \(\lim\)_{{x \(\to\) 1^-}} 2x.
4. Step 4: Calculate the right-hand limit as x approaches 1 from the right (x \(\to\) 1^+). Use the expression for x \(\geq\) 1, which is x^2 + 3x. Evaluate the limit: \(\lim\)_{{x \(\to\) 1^+}} (x^2 + 3x).
5. Step 5: Check the value of the function at x = 1. Since x = 1 falls into the second piece of the function (x \(\geq\) 1), calculate f(1) using x^2 + 3x. Compare this value with the left-hand and right-hand limits to determine if the function is continuous at x = 1.

Key Concepts

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Understanding continuity is essential for determining where a function behaves predictably without breaks, jumps, or asymptotes.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, the function f(x) has two distinct expressions: one for x < 1 and another for x ≥ 1. Analyzing piecewise functions requires checking the continuity at the boundaries where the pieces meet, which is crucial for determining the overall continuity of the function.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. To assess continuity at the boundary of a piecewise function, one must evaluate the left-hand limit and the right-hand limit at that point. If both limits exist and are equal to the function's value at that point, the function is continuous there.