How many discontinuities does the following piecewise function have?

Question

math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row 3 e superscript x plus 3 baseline plus 1 comma x less than negative 3 2nd row start fraction 2 over 3 end fraction x squared minus 1 comma negative 3 less than x less than or equals 3 3rd row negative start fraction 7 over 2 end fraction x plus start fraction 29 over 2 end fraction comma 3 less than x less than 5 4th row log left parenthesis 2 x minus 4 right parenthesis comma x greater than or equals 5 end layout
(1 point)
Responses

three
three

two
two

one
one

zero

Answer 1

To determine the number of discontinuities in the given piecewise function, we need to check the points at which the function changes its definition, specifically at $x=-3$, $x=3$, and $x=5$. We will assess the continuity of the function at those points.

1. At $x=-3$:

   • From the left ($x<-3$): $f(x)=3e^x+1$
     • $f(-3)=3e^{-3}+1$ (a specific value).
   • From the right ($-3<x\le 3$): $f(x)=\frac{2}{3}{x}^2-1$
     • $f(-3)=\frac{2}{3}(-3{)}^2-1=\frac{2}{3}\cdot 9-1=6-1=5$.
   • Since $3e^{-3}+1\ne 5$, there is a discontinuity at $x=-3$.

2. At $x=3$:

   • From the left ($-3<x\le 3$): $f(x)=\frac{2}{3}{x}^2-1$
     • $f(3)=\frac{2}{3}(3{)}^2-1=\frac{2}{3}\cdot 9-1=6-1=5$.
   • From the right ($3<x<5$): $f(x)=-\frac{7}{2}x+\frac{29}{2}$
     • $f(3)=-\frac{7}{2}(3)+\frac{29}{2}=-\frac{21}{2}+\frac{29}{2}=\frac{8}{2}=4$.
   • Since $5\ne 4$, there is a discontinuity at $x=3$.

3. At $x=5$:

   • From the left ($3<x<5$): $f(x)=-\frac{7}{2}x+\frac{29}{2}$
     • $f(5)=-\frac{7}{2}(5)+\frac{29}{2}=-\frac{35}{2}+\frac{29}{2}=-\frac{6}{2}=-3$.
   • From the right ($x\ge 5$): $f(x)=\log (2x-4)$
     • $f(5)=\log (2\cdot 5-4)=\log (10-4)=\log (6)$.
   • Since $-3\ne \log (6)$, there is a discontinuity at $x=5$.

Summary:

• The function has discontinuities at $x=-3$, $x=3$, and $x=5$.

Thus, the total number of discontinuities is 3. The correct response is:

three