Consider the graph of y = f \left( x \right), which is defined for all real x:
Define the function for x > 4.
Define the function for x \leq 4.
Consider the graph of y = f \left( x \right):
Is the function defined for all x?
Define the function for x > 4.
Define the function for x \leq 4.
Is the graph continuous at x = 4?
Consider the graph of y = f \left( x \right):
Find the function value at x = 3.
Is the graph continuous at x = 3?
Consider the graph of y = f \left( x \right):
Find the function value at x = 2.
Is the graph continuous at x = 2?
Consider the graph of y = f \left( x \right):
Find the function value at x = 4.
Is the graph continuous at x = 4?
Consider the graph of y = f \left( x \right):
Find the function value at x = - 5.
Is the graph continuous at x = - 5?
Consider the graph of y = f \left( x \right):
Is the function defined for all real values of x?
Complete the following to define the function:
f\left(x\right) = \begin{cases} ⬚, & \enspace ⬚\leq x\leq ⬚ \\ x +3, & \enspace x \gt 4 \\ \end{cases}Is the graph continuous at x = 4?
Consider the graph of y = f \left( x \right):
Is the function defined for all real values of x?
Define the function for x > 2.
Define the function for x < 2.
Is the graph continuous at x = 2?
Define the piecewise functions in the graphs below:
Sketch the following piecewise functions:
For each of the following piecewise functions:
Sketch the function.
State the values of x for which f(x) is defined.
State the values of x for which f(x) is continuous.
Consider the following piecewise relationship: f\left(x\right) = \begin{cases} -4 x - 6, & x \leq a \\ x - 1, & x \gt a \end{cases}
If the graph of the piecewise relationship is connected, solve the equations simultaneously to find the value of a.
Find the value of f(a).
Consider the following piecewise function:
f\left(x\right) = \begin{cases} -4-x, & \enspace x \leq a \\ x-1, & \enspace x \gt a \end{cases}If the graph of the piecewise function is continuous, find the value of a.
Find the value of f(a).
A kitchen sink starts with 20 \text{ L} of water. It empties at a rate of 4 \text{ L} per minute and then the drain gets blocked. No water empties for 2 more minutes while it gets unblocked and then the remaining water is drained out in 1 minute.
Construct the piecewise graph that represents the above scenario.
A tank is being filled at a constant rate of 4 \text{ L} per second for 4 minutes. The hose is then it is turned off for 2 minutes. Finally it is topped up again with water at a rate of 3 \text{ L} per second for 1 minute.
Construct the piecewise graph that represents the above scenario.
The graph below shows Ben's distance from his starting point at various times of the day using 24-hour time:
At what time did Ben start his journey?
How far had Ben travelled by 11 am?
What happened to Ben's speed at 11 am?
Calculate Ben's speed between 11 am and 1 pm.
What distance did Ben travel between 1 pm and 2 pm?
What is the furthest distance travelled from the starting point?
Calculate the total distance travelled by Ben from 9 am to 4 pm.
Ken starts travelling at 9 am from point A to point B. The distance between Ken and point B at various times in his journey is shown on the graph below:
How far is point B from point A?
How many hours was Ken's journey from point A to point B?
State the time period in which Ken travelled at the fastest speed.
Find Ken's fastest speed in \text{km/h}.
Find the distance in kilometres travelled by Ken between 1:30 pm and 4 pm.
Find the total distance travelled by Ken from 9 am to 4 pm.
A children's pool is being filled with water. The volume of water V, in litres, after t minutes is given by the following piecewise graph. The pool has a maximum capacity of 1500 \text{ L}.
Estimate the volume of water in litres after 45 minutes have elapsed, to the nearest 10 litres.
Determine the equation that describes the volume of water V in litres after t minutes in the first 45 minutes.
After 45 minutes, the rate at which the volume of water enters the pool is increased. The piecewise relationship after t = 45 is given by V = 26 t - 970.
Use the equation to determine the value of t (in minutes) when the pool is filled up.
A marathon runner initially runs at an average speed of 15 \text{ km/h}. After 40 minutes of running at this speed, the runner slows down to a more comfortable speed of 12 \text{ km/h} for the remaining 160 minutes.
Let D be the distance covered by the marathon runner in kilometres and t be the time elapsed in minutes.
Define piecewise relationship D(t).
Find the distance covered in kilometres after the following number of minutes have elapsed:
20 minutes
40 minutes
60 minutes
200 minutes
Sketch the graph of the piecewise function D(t).
Harry goes out for a run. He accelerates from rest up to a desired speed and maintains that speed for some time. Feeling exhausted, his speed drops until he's back at rest.
The speed, S \text{ km/h}, after t seconds is given by the following piecewise relationship:
S = \begin{cases} 0.8t, & \enspace 0\leq t \lt 15 \\ 12, &\enspace 15\leq t \lt 240 \\ 252-t, & \enspace 240\leq t \leq 252 \end{cases}Find the speed after the following number of seconds have elapsed:
10 seconds
50 seconds
150 seconds
250 seconds
Sketch the graph of this piecewise relationship.
The amount of energy stored, E \text{ kWh}, in a set of solar batteries t hours after 6:00 am is given by the following piecewise graph. The storage is shown for a 36-hour period.
Part of the piecewise graph is given by: E = \dfrac{7}{6} t, \enspace 0 \leq t \leq a
Determine the value of a.
Determine the equation for the horizontal part of the piecewise graph.
The equation that describes the last piece of the piecewise graph is \\ E = t - 10.
State the time interval that the last piece is defined over.
A real estate agent sells a property and earns commission at the following rates:
1.5\% on the first \$20\,000
0.9\% on the remainder
Calculate the commission earned on sales of:
Define the piecewise function for the commission, C.
Sketch the graph of the piecewise function.
A real estate agent sells a property and earns commission at the following rates:
3.7\% on the first \$50\,000
1.2\% on the next \$25\,000
0.8\% on the remainder
Calculate the commission earned on sales of:
Define the piecewise function for the commission, C.
Sketch the graph of the piecewise function.
Consider the 1999-2000 tax table below:
| Taxable Income | Tax on this income |
|---|---|
| \$1-\$5400 | \text{Nil} |
| \$5401-\$20\,700 | 20 \text{ cents for each }\$1\text{ over }\$5400 |
| \$20\,701-\$38\,000 | \$3060 + 34 \text{ cents for each }\$1\text{ over }\$20\,700 |
| \$38\,001-\$50\,000 | \$8942 + 43 \text{ cents for each }\$1\text{ over }\$38\,000 |
| \$50\,001\text{ and over} | \$14\,102 + 47 \text{ cents for each }\$1\text{ over }\$50\,000 |
Calculate the tax payable on an income of:
Write the piecewise function defining this income tax table.
Sketch the graph of the piecewise function.
Consider the 2019-2020 tax table below:
| Taxable Income | Tax on this income |
|---|---|
| 0-\$18\,200 | \text{Nil} |
| \$18\,201-\$37\,000 | 19 \text{ cents for each }\$1\text{ over }\$18\,200 |
| \$37\,001-\$90\,000 | \$3572 + 32.5 \text{ cents for each }\$1\text{ over }\$37\,000 |
| \$90\,001-\$180\,000 | \$20\,797 + 37 \text{ cents for each }\$1\text{ over }\$90\,000 |
| \$180\,001\text{ and over} | \$54\,097 + 45 \text{ cents for each }\$1\text{ over }\$180\,000 |
Calculate the tax payable on an income of:
Write the piecewise function defining this income tax table.
Sketch the graph of the piecewise function.
An insurance company charges a fixed monthly premium depending on which age group you fall into.
The piecewise function outlines the premium, y, for a given age, x:
y = \begin{cases} \$112.17, & 18\lt x \leq 25 \\ \$128.72, & 25 \lt x \leq 35 \\ \$151.73, & 35 \lt x \leq 45 \\ \$176.27, & 45 \lt x \leq 55 \\ \$184.52, & 55 \lt x \leq 65 \\ \end{cases}
How much will the monthly premium for a 22-year old person be?
What could be the maximum age of someone who is paying \$176.27 per month for their premium?
Fikile held an insurance policy from when she turned 26 to when she turned 30. During this time, how much did she pay altogether in premiums?
If you are 35 years old, how much will your annual premium increase by once you turn 36 years old?
The government is looking at alternative taxation schemes and one proposal is to pay a fixed amount of tax depending on which income bracket you fall into.
The given piecewise function models the tax payable, y, based on an income of x:
y = \begin{cases} \$1200, & \$0 \lt x \leq \$15\,000 \\ \$4900, & \$15\,000 \lt x \leq \$35\,000 \\ \$12\,350, & \$35\,000 \lt x \leq \$65\,000 \\ \$43\,320, & \$65\,000 \lt x \leq \$114\,000 \\ \$54\,720, & x > \$114\,000 \\ \end{cases}
Under this proposal, how much tax will you pay on an income of \$14\,000?
Determine the difference in income between the highest income earner and lowest income earner who are both paying \$43\,320 in tax. Assume you are taxed on whole dollars you earn.
Estelle is earning an income of \$63\,800. She is set to get a pay rise at the end of the year. Find the maximum pay rise that she can get before falling into the next highest tax bracket.
At an indoor ski facility, the temperature is set to - 5 \degree \text{C} at 2 pm. At 3 pm, the temperature is immediately brought down to - 12 \degree \text{C} and left for 3 hours before immediately taking it down again to - 18 \degree \text{C}, where it stays for the rest of the day’s operation. The facility operates until 10 pm.
Write a stepwise function that models the indoor temperature, y, at a certain time of the day, x hours after midday.
Sketch a graph of the step function relating time of day and temperature inside the ski facility.
Lakota entered the ski facility at 3:30 pm. What was the temperature inside the facility at this time?
Xavier wants to wait till the indoor temperature is - 7 \degree \text{C} or lower. When is the earliest he can enter the facility?