Q. 80 In Exercises 69鈥�80, determine ... [FREE SOLUTION]

Chapter 2: Q. 80 (page 185)

In Exercises 69鈥�80, determine whether or not f is continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.
$f (x) = \{\begin{cases} x^{2}, & if x rational \\ 2 x - 1, & if x irrational \end{cases}, x = 1$

Short Answer

Expert verified

The function $f (x) = \{\begin{cases} x^{2}, & if x rational \\ 2 x - 1, & if x irrational \end{cases}$ is continuous and differentiable at localid="1648501951781" $x = 1 .$

Step by step solution

Step 1. Given information.

The given function is $f (x) = \{\begin{cases} x^{2}, & if x rational \\ 2 x - 1, & if x irrational \end{cases} .$

The given value of x is $x = 1 .$

Step 2. Graph of function.

Plot the graph of the function.

Step 3. Continuity of a function.

Graph of function $f (x) = x^{2}$ state that the value of the function is approaching to $1 .$

Graph of function $f (x) = 2 x - 1$ state that the value of the function is approaching to $1 .$

The value of the function $f (x) = 1$ at $x = 1 .$

The point of intersection of both functions is role="math" localid="1648501599030" $(1, 1) .$

so the function $f (x)$ is continuous at $x = 1 .$

Step 4. Differentiability of function.

Differentiate the function

$f (x) = \{\begin{cases} x^{2}, & if x rational \\ 2 x - 1, & if x irrational \end{cases} f' (x) = \{\begin{cases} 2 x, & if x rational \\ 2, & if x irrational \end{cases}$