   Chapter 2.5, Problem 42E

Chapter
Section
Textbook Problem

Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. f ( x ) = { 2 x   if   x < 1 3 − x       if   1 < x ≤ 4 x   if   x > 4

To determine

To find: The function f(x)={2xif x13xif 1<x4xif x>4 is discontinuous at which numbers and explain for which of the numbers are continuous from the right, from the left, or neither. Sketch the graph of the function f(x)={2xif x13xif 1<x4xif x>4.

Explanation

Definition used: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Note 1: “If f is defined near a, f is discontinuous at a whenever f is not continuous at a”.

Theorem used:

1. The functions such as “Polynomials, rational functions, root functions, trigonometric functions, inverse trigonometric functions, exponential functions and logarithmic functions” are continuous at every number in their domains.

2. A function f is continuous from the right at a number a if limxa+f(x)=f(a) and a function f is continuous from the left at a number a if limxaf(x)=f(a).

3. The limit limxaf(x)=L if and only if limxaf(x)=L=limxa+f(x).

4. If f is continuous at b and limxag(x)=b, then limxaf(g(x))=f(limxag(x)).

Calculation:

By note 1, the function f is said to be discontinuous at x=a if anyone of the following conditions does not satisfied.

• f(a) is defined
• The limit of the function at the number a exists.
• limxaf(x)=f(a)

Consider the piecewise function f(x)={2xif x13xif 1<x4xif x>4 .

Here, the function f(x)=2x is an exponential function defined in the interval (,1), f(x)=3x is a polynomial defined in the interval (1,4) and f(x)=1x is a rational function defined in the interval (4,).

Since f(x)=2x is a exponential function, f(x)=3x is a polynomial function and f(x)=x is a root function and by theorem 1, those functions are continuous on its respective domains.

Therefore, f is continuous on the interval (,1)(1,4)(4,).

So that, f might be discontinuous at 1 and 4.

Check the discontinuity of f at x=1.

At x=1, then f(1)=2 is defined. (1)

The limit of the function f(x) as x approaches a=1 is computed as follows

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