Question

2x-1, x < 2 12. Show that f(x) = { 3x 2 x ≥ 2 is continuous.​

Answer 1

Using the continuity concept, since the lateral limits and the numeric value of the function are equal at the point in which the definition changes, the function is continuous.

What is the continuity concept?

A function f(x) is continuous at x = a if it is defined at x = a, and:

$\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)$

The definition of the piecewise function is given by:

• f(x) = 2x - 1, x < 2.

• f(x) = 3x/2, x >= 2.

Since the definition of the function changes at x = 2, and the domain of the function has no restrictions, this is the only point in which there may be a discontinuity.

The lateral limits are:

• $\lim_{x \rightarrow 2^-} f(x) = \lim_{x \rightarrow 2} 2x - 1 = 2(2) - 1 = 3$.

• $\lim_{x \rightarrow 2^+} f(x) = \lim_{x \rightarrow 2} 1.5x = 1.5(2) = 3$.

The numeric value is:

f(2) = 1.5 x 2 = 3.

Since the lateral limits and the numeric value of the function are equal at the point in which the definition changes, the function is continuous.

More can be learned about the continuity concept at brainly.com/question/24637240

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