Question

Is​ f(x) continuous at x equals 4​? Why or why​ not? A. ​No, f(x) is not continuous at x equals 4 because ModifyingBelow lim With x right arrow 4 f (x )does not exist. B. ​Yes, f(x) is continuous at x equals 4 because f (4 )exists. C. ​No, f(x) is not continuous at x equals 4 because f (4 )is undefined. D. ​Yes, f(x) is continuous at x equals 4 because ModifyingBelow lim With x right arrow 4 f (x )equals f (4 ).

Answer 1

Corrected Question

Is the function given by:

$f(x)=\left\{\begin{array}{ccc}\frac{1}{4}x+1 &x\leq 4\\4x-11&x>4\end{array}\right$ ​

continuous at x=4​? Why or why​ not? Choose the correct answer below.

Answer:

(D) ​Yes, f(x) is continuous at x = 4 because $Lim_{x \to 4}f(x)=f(4)$

Step-by-step explanation:

Given the function:

$f(x)=\left\{\begin{array}{ccc}\frac{1}{4}x+1 &x\leq 4\\4x-11&x>4\end{array}\right$

A function to be continuous  at some value c in its domain if the following condition holds:

• f(c) exists and is defined.

• $Lim_{x \to c} f(x)$ exists.

• $f(c)=Lim_{x \to c} f(x)$

At x=4

• $f(4)=\dfrac{1}{4}*4+1=2$

• $Lim_{x \to 4}f(x)=2$

Therefore: $Lim_{x \to 4}f(x)=f(4)=2$

By the above, the function satisfies the condition for continuity.

The correct option is D.