Piecewise functions are functions that have multiple pieces, or sections. They are defined piece by piece, with various functions defining each interval.
Piecewise functions can be split into as many pieces as necessary. Each piece behaves differently based on the input function for that interval. Pieces may be single points, lines, or curves. The piecewise function below has three pieces. The piece on the interval represents the function The piece on the interval represents the function The piece on the interval represents the function
Using function notation, we represent the graph as:
A certain cab company has a $2.00 base charge, and then charges $0.50 per minute. There is also a $7.00 minimum fee (so if the base charge and minutes combined don't add to $7, the rider is charged a flat amount of $7).
Which function describes riding a cab from the company for minutes and spending dollars?
Remember that is the number of minutes, but the cutoff for the minimum fare is based on dollars. So we need to figure out how many minutes will reach $7. That would be when so or This means we want the piecewise function to be split starting at 10 minutes, when the minimum fare threshold is passed.
One of the most common piecewise functions is the absolute value function. How can we write as a piecewise function?
is the combination of two linear functions:
When evaluating a piecewise function, we need to determine which piece of the function to use. Let's find if
indicates that we want to determine the value of the function when An -value of falls into the first piece of the function, where for Therefore,
f\(3 falls into the piece of the graph where for Therefore,
If piecewise function given below is continuous, then what is the value of (In the context of this problem, "continuous" means the endpoints of the graph portions meet at so there is no "gap".)
At the graph is at the point
When and the graph is at the point
So we need to shift the parabola graph down by so the points match. This indicates
To graph a piecewise function, we graph the different pieces for the different sub-intervals. Let's graph
This piecewise graph has three pieces and two boundary points at and The first piece of our graph is the linear function for so we'll have a filled in dot at with a slope of 2 traveling from the point toward negative infinity.
Next, we have the quadratic function for with boundary points of and so we'll have an open dot at and so we'll have a closed dot at .
The third piece is the horizontal linear function of from to infinity.
What is the correct graph of
Graph A has the correct functions but the wrong boundary point of instead of Graph C has the correction functions and the correct boundary point, but the dot should be a closed dot because the first function includes the value Therefore, Graph B is correct.