5.04 Domain and range
For each of the following relations:
Find the domain.
Find the range.
Determine whether the relation is a function or not.
\left\{\left(8, 5\right), \left(1, 3\right), \left(3, 4\right), \left(9, 1\right), \left(2, 9\right)\right\}
\left\{\left(2, 3\right), \left(3, 8\right), \left(6, 1\right), \left(8, 7\right), \left(3, 2\right)\right\}
\left\{\left(4, 4\right), \left(8, 9\right), \left(2, 8\right), \left(7, 9\right), \left(3, 1\right)\right\}
| x | 1 | 6 | 3 | 8 | 2 |
|---|---|---|---|---|---|
| y | 3 | 2 | 7 | 1 | 2 |
| x | 7 | 7 | 8 | 5 | 3 |
|---|---|---|---|---|---|
| y | 1 | 9 | 3 | 2 | 6 |
State the domain and range of the following functions:
Consider the function f graphed. Use the graph to determine whether each of the given statements are true or false:
The domain is \left[ - 5 , 7\right]
The range is \left[ - 3 , 16\right]
f \left( 1 \right) - f \left( 7 \right) = 16
f \left( 0 \right) = 16
What is the domain of the following functions?
f \left( x \right) = - 3 x - 1
f \left( x \right) = 8 x + 8
The graph of y = 2^{x} is shown below:
What is the y-intercept of this graph?
Does the graph have an x-intercept?
What is the graph's domain?
What is the graph's range?
Find the value of y when x = 7.
Find the value of x when y = 256.
Consider the graph of the function shown:
What is the maximum value of the graph?
Hence, determine the range of the function.
What is the domain of this function?
Consider the graph of the function shown:
What is the minimum value of the graph?
Hence, determine the range of the function.
Over what interval of the domain is the function increasing?
For the following quadratic functions:
| x | y |
|---|---|
| 1 | 17 |
| 2 | 7 |
| 3 | 1 |
| 4 | -1 |
| 5 | 1 |
| 6 | 7 |
| 7 | 17 |
| x | y |
|---|---|
| -5.5 | -12 |
| -5 | -7 |
| -4.5 | -4 |
| -4 | -3 |
| -3.5 | -4 |
| -3 | -7 |
| -2.5 | -12 |
Consider the graph of the function f \left(x\right) = \sqrt{x + 1} below:
State the domain of the function.
Is there a value in the domain that can produce a function value of -2?
Consider the parabola defined by the equation y = x^{2} + 5.
Is the parabola concave up or concave down?
What is the minimum y value of the parabola?
Hence, determine the range of the parabola.
For the following functions:
f \left( x \right) = - x + 1
f \left( x \right) = 3 x - 1
f \left( x \right) = - \dfrac{3}{2} x - 2
f \left( x \right) = - 4
x = - 1
Consider the function y = x^{3}-1.
Complete the table of values:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function.
Is there a value in the domain that can produce a function value of 3 \dfrac{2}{3} ?
Consider f \left(x\right) = x + 3 for the domain \{- 5, -4, 0, 1\}.
Complete the table of values:
| x | -5 | -4 | 0 | 1 |
|---|---|---|---|---|
| f(x) |
Plot the points on a number plane.
Consider f \left(x\right) = - 2 x for the domain \{- 1, 0, 1, 2\}.
Complete the table of values:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| f(x) |
Plot the points on a number plane.
Consider the function f \left(x\right) = \dfrac{x + 3}{2}, for all real x.
Complete the table of values:
| x | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function.
Consider - 2 x + y = 1 for the domain \{- 4, -3, 0, 2\}.
Make y the subject of the equation.
Complete the table of values:
| x | -4 | -3 | 0 | 2 |
|---|---|---|---|---|
| y |
Plot the points on a number plane.
Consider f \left(x\right) = x^{2} + 4 for the domain \{-3, -1, 0, 1, 3\}.
Complete the table of values:
| x | -3 | -1 | 0 | 1 | 3 |
|---|---|---|---|---|---|
| f(x) |
Plot the points on a number plane.
Consider f \left(x\right) = \left(x + 2\right)^{2} for the domain \{- 2, -1, 0, 1, 2\}.
Complete the table of values:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| f(x) |
Plot the points on a number plane.
The function f is used to determine the area of a square given its side length.
State whether the following values are part of the domain of the function:
- 8
6.5
9\dfrac{1}{3}
\sqrt{78}
For n \geq 0, state the area function for a side length of n.
Sketch the graph of the function f.
What is the domain of the function defined by f \left( x \right) = \dfrac{1}{x + 5} ?
For each of the following graphs of rational functions:
State the domain using interval notation.
State the range using interval notation.
An industrial process involves heating a material from 25°C to 80°C in order to remove impurities.
Using the standard approach, the temperature of the material over time is given by the function: T = -t \left(t - 16\right) + 25
A graph of the temperature is given, where temperature T is given in degrees Celsius(°C) and time t is in minutes.
What is the domain of this function?
What is the range of this function?
A new heating process is being developed that uses more energy but takes less time. This regime is based on a different quadratic function, and the table below shows the theoretical value of the temperature at different points in time using the new function.
| t | 0 | 1 | 3 | 4 | 30 | 31 | 32 | 33 | 34 | 35 |
|---|---|---|---|---|---|---|---|---|---|---|
| T | 25 | 45 | 80 | 96 | 96 | 80 | 63 | 45 | 25 | 4 |
For what values of t does this new function have the same range as the standard process?
Hence, what values of t should be used for the new heating process?
The number of bees in a colony is initially measured and left to form a hive. The number of bees in the hive is measured each day over the course of one week. The function
H \left(x\right) = 30 \left(3\right)^{x} is found to model the number of bees, H, after x days.
What is the domain of the function?
What is the initial number of bees in the colony?
Can the function be used to model the number of bees over an entire season?
At an indoor ski facility, the temperature is set to - 8°C at the opening time of 9 am. At 10 am, the temperature is immediately brought down to - 15°C and left for 3 hours before immediately taking it down again to - 23 °C where it stays until the close time of 7 pm.
Write the piecewise function that models the indoor temperature f\left(x\right) in terms of the number of hours after the facility has opened, x.
The graph of the function is given. State the domain of the function.
Kate entered the ski facility at 2:30 pm. What was the temperature inside the facility?
Vincent wants to wait until the indoor temperature is - 10\degree C or lower. When is the earliest he can enter the facility?