Piecewise function

[kickingtoad]

A long-distance telephone service charges $ 0.56 a minute for the first 30 minutes or less of a call and $ 0.06 per minute for each additional minute or fraction thereof.

(A) Complete the piecewise definition of the charge F(x) for a long-distance call lasting x minutes.

\(\displaystyle \;f(x) \;=\;\begin{Bmatrix}S(x) && \text{if }x\,\le a \\ \\[-3mm] L(x) && \text{if }x> a \end{array}\)

Where
\(\displaystyle S(x)=.56x\)
\(\displaystyle L(x)=?\) I was thinking (.56x30)+.06x
\(\displaystyle a=30\)

\(\displaystyle \lim_{x\to\ 30-}=?\)

\(\displaystyle \lim_{x\to\ 30+}=?\)

\(\displaystyle \lim_{x\to\ 30}=?\)

 

[DrSteve]

It would be (.56x30) + .06(x - 30) if it weren't for the "fraction thereof" part.

You need to throw in the ceiling function so that you always get an integer.

 

[mmm4444bot]

Do you have a question about limits?

 

[kickingtoad]

I figured it out. The limit as x approaches 0 is 16.8 because of .56x ==> .56(30)=16.8

 

[lookagain]

A long-distance telephone service charges $ 0.56 a minute for the first
30 minutes or less of a call and $ 0.06 per minute for each additional minute or fraction thereof.

(A) Complete the piecewise definition of the charge F(x) for a long-distance call lasting x minutes.

\(\displaystyle \;f(x) \;=\;\begin{Bmatrix}S(x) && \text{if }x\,\le a \\ \\[-3mm] L(x) && \text{if }x> a \end{array}\)

\(\displaystyle where\)

\(\displaystyle S(x) \ =\ .56x\)

\(\displaystyle L(x) \ = \ ? \ \ \ \ \ \ \ I \ was \ thinking\ >>> \ \ \\)(.56x30) + .06x \(\displaystyle \ \ \ <<<\)

kickingtoad,

for your part highlighted here, do not type a multiplication sign that is identical looking
to the variable(s), which in your case is "x." \(\displaystyle \ \longleftarrow[Fact] \\)

In fact, because you are working in algebra, it is advisable not to type the times sign,
especially with variables. \(\displaystyle \ \longleftarrow [Opinion, \ because \ of \ style/readability].\)\(\displaystyle **\)

As a suggestion, type it as:

.56(30) + .06x or

\(\displaystyle .56(30) \ + \ .06x\)



\(\displaystyle **\)

\(\displaystyle Three \ examples \ of \ awkward-looking \ uses \ of \ the \ times \ sign \ with \ variables:\)


\(\displaystyle 1. \ \ \ X \times X \times X \times X \ = \ X^4\)


\(\displaystyle 2. \ \ \ x \times x \times \\ x \ = \ x^3\)


\(\displaystyle 3. \ \ \ w \times y \times x \times y \times y \times z \ = \ w \times x \times y \times y \times y \times z \ = \ wxy^3z\)