suppose that the function f is defined for all real numbers as follows.\n$f(x)=\\begin{cases}1 +…

suppose that the function f is defined for all real numbers as follows.\n$f(x)=\\begin{cases}1 + x^{2}&\\text{if }x < 1\\\\2x + 3&\\text{if }x\\geq1\\end{cases}$\ngraph the function f. then determine whether or not the function is continuous.\nis the function f continuous?\nyes\nno

suppose that the function f is defined for all real numbers as follows.\n$f(x)=\\begin{cases}1 + x^{2}&\\text{if }x < 1\\\\2x + 3&\\text{if }x\\geq1\\end{cases}$\ngraph the function f. then determine whether or not the function is continuous.\nis the function f continuous?\nyes\nno

Answer

Explanation:

Step1: Find ( f(1) )

Since ( x = 1) and (x\geq1), use (f(x)=2x + 3). (f(1)=2\times1+3=5)

Step2: Find (\lim_{x\rightarrow1^{-}}f(x))

As (x\rightarrow1^{-}) (approaching 1 from the left, (x < 1)), use (f(x)=1+x^{2}). (\lim_{x\rightarrow1^{-}}f(x)=\lim_{x\rightarrow1^{-}}(1 + x^{2})=1+1^{2}=2)

Step3: Find (\lim_{x\rightarrow1^{+}}f(x))

As (x\rightarrow1^{+}) (approaching 1 from the right, (x\geq1)), use (f(x)=2x + 3). (\lim_{x\rightarrow1^{+}}f(x)=\lim_{x\rightarrow1^{+}}(2x + 3)=2\times1+3=5)

Step4: Check the continuity condition

A function (y = f(x)) is continuous at (x = a) if (\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)=f(a)). Here, (\lim_{x\rightarrow1^{-}}f(x)=2\neq f(1) = 5)

Answer:

No