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Activity2.13Piecewise-Defined Functions

Piecewise-defined functions are functions where the formula used depends upon the value of the input. When looking for discontinuities on piecewise-defined functions, you need to investigate the behavior at values where the formula changes as well as values where the issues discussed in Activity 2.12 might pop up.

Subsection2.13.1Exercises

This question is all about the function \(f\) defined by \begin{equation*}\fe{f}{x}=\begin{cases}\frac{4}{5-x}&x\lt1\\\frac{x-3}{x-3}&1\lt x\lt4\\2x+1&4\leq x\leq7\\\frac{15}{8-x}&x\gt7\text{.}\end{cases}\end{equation*}

1

Complete Table 2.13.1.

\(a\) \(\fe{f}{a}\) \(\lim\limits_{x\to a^{-}}\fe{f}{x}\) \(\lim\limits_{x\to a^{+}}\fe{f}{x}\) \(\lim\limits_{x\to a}\fe{f}{x}\)
\(1\)
\(3\)
\(4\)
\(5\)
\(7\)
\(8\)
Table2.13.1Function values and limit values for \(f\)
2

Complete Table 2.13.2.

Location of
discontinuity
From Definition 2.9.1,
sub-property (1, 2, 3)
that is not met
Removable? Continuous
from one side?
\(\strut\)
\(\strut\)
\(\strut\)
\(\strut\)
Table2.13.2Discontinuity analysis for \(f\)

Consider the function \(f\) defined by \(\fe{f}{x}=\begin{cases}\frac{5}{x-10}&x\leq5\\\frac{5}{5x-30}&5\lt x\lt7\\\frac{x-2}{12-x}&x>7\text{.}\end{cases}\)

State the values of \(x\) where each of the following occur. If a stated property doesn't occur, make sure that you state that (as opposed to simply not responding to the question). No explanation necessary.

3

At what values of \(x\) is \(f\) discontinuous?

4

At what values of \(x\) is \(f\) continuous from the left, but not from the right?

5

At what values of \(x\) is \(f\) continuous from the right, but not from the left?

6

At what values of \(x\) does \(f\) have removable discontinuities?

Consider the function \(g\) defined by \(\fe{g}{x}=\begin{cases}\frac{C}{x-17}&x\lt10\\C+3x&x=10\\2C-4&x\gt10\text{.}\end{cases}\)

The symbol \(C\) represents the same real number in each of the piecewise formulas.

7

Find the value for \(C\) that makes the function continuous on \(\ocinterval{-\infty}{10}\). Make sure that your reasoning is clear.

8

Is it possible to find a value for \(C\) that makes the function continuous over \(\ointerval{-\infty}{\infty}\)? Explain.