Determine where the function
$\text{\hspace{0.17em}}f(x)=\{\begin{array}{l}\frac{\pi x}{4},\text{}x2\\ \frac{\pi}{x},\text{}2\le x\le 6\\ 2\pi x,\text{}x6\end{array}\text{\hspace{0.17em}}$ is discontinuous.
To determine whether a
piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.
Given a piecewise function, determine whether it is continuous.
Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
For each boundary point
$\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ of the piecewise function, determine if each of the three conditions hold.
Determining whether a piecewise function is continuous
Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.
The two functions composing this piecewise function are
$\text{\hspace{0.17em}}f(x)=\mathrm{sin}(x)\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}x<0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}f(x)={x}^{3}\text{\hspace{0.17em}}$ on
$\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,
At
$\text{\hspace{0.17em}}x=0,$ let us check the three conditions of continuity.
Because all three conditions are not satisfied at
$\text{\hspace{0.17em}}x=0,$ the function
$\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ is discontinuous at
$\text{\hspace{0.17em}}x=0.$
A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See
[link] .
Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See
[link] and
[link] .
For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See
[link] and
[link] .
Section exercises
Verbal
State in your own words what it means for a function
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ to be continuous at
$\text{\hspace{0.17em}}x=c.$
Informally, if a function is continuous at
$\text{\hspace{0.17em}}x=c,$ then there is no break in the graph of the function at
$\text{\hspace{0.17em}}f\left(c\right),$ and
$\text{\hspace{0.17em}}f\left(c\right)\text{\hspace{0.17em}}$ is defined.
For the following exercises, determine why the function
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is discontinuous at a given point
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ on the graph. State which condition fails.
Discontinuous at
$\text{\hspace{0.17em}}a=3$ ;
$\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}f(x)=3,$ but
$\text{\hspace{0.17em}}f(3)=6,$ which is not equal to the limit.
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.