−− ++ −− ++(d) The solution set of the inequality is given by
Lesson 6 Supplementary Exercises Lesson 6 Supplementary Exercises
1. Solve for x:
2. Solve for x:
3. Solve for x:
4. Solve for x:
5. Solve for x:
6. If a and b are real numbers such that
, find the solution set of
.7. You have 6 liters of a pineapple juice blend that has 50% pure pineapple juice.
How many liters of pure pineapple juice needs to be added to make a juice blend that is 75% pineapple juice?
8. Two ships traveling from Dumaguete to Cagayan de Oro differ in average speed by 10 kph. The slower ship takes 3 hours longer to travel a 240-kilometer route than for the faster ship to travel a 200-kilometer route. Find the speed of the slower ship.
D E P E D C O P Y
Lesson 7: Representations of Rational Functions Lesson 7: Representations of Rational Functions
Learning Outcome(s)
Learning Outcome(s): At the end of the lesson, the learner is able to represent a rational function through its table of values, graphs and equation, and solve problems involving rational functions.
Lesson Outline:
Lesson Outline:
1. Table of values, graphs and equations as representations of a rational function.
2. Rational functions as representations of real-life situations
Definition:
Definition: A rational functionrational function is a function of the form
where
and
are polynomial functions and
is not the zero polynomial (i.e.,
). The domain of
is all values of
where
.Average speed (or velocity) can be computed by the formula
. Consider a 100-meter track used for foot races. The speed of a runner can be computed by taking the time for him to run the track and applying it to the formula
,since the distance is fixed at 100 meters.
Example 1.
Example 1. Represent the speed of a runner as a function of the time it takes to run 100 meters in the track.
Solution. Since the speed of a runner depends on the time it takes to run 100 meters, we can represent speed as a function of time.
Let x represent the time it takes to run 100 meters. Then the speed can be represented as a function
as follows:
Observe that it is similar to the structure to the formula
relating speed, distance, and time.Example 2.
Example 2. Continuing the scenario above, construct a table of values for the speed of a runner against different run times.
Solution. A table of values can help us determine the behavior of a function as the variable
changes.The current world record (as of October 2015) for the 100-meter dash is 9.58 seconds set by the Jamaican Usain Bolt in 2009. We start our table of values at 10 seconds.
D E P E D C O P Y
Let x be the runtime and
be the speed of the runner in meters per second, where
. The table of values for run times from 10 seconds to 20 seconds is as follows:
10 12 14 16 18 20
10 8.33 7.14 6.25 5.56 5From the table we can observe that the speed decreases with time. We can use a graph to determine if the points on the function follow a smooth curve or a straight line.
Example 3.
Example 3. Plot the points on the table of values on a Cartesian plane.
Determine if the points on the function
follow a smooth curve or a straight line.Solution. Assign points on the Cartesian plane for each entry on the table of values above:
A(10,10) B(12,8.33) C(14, 7.14) D(16, 6.25) E(18,5.56) F(20,5)
Plot these points on the Cartesian plane:
By connecting the points, we can see that they are not collinear but rather follows a smooth curve.
D
For the 100-meter dash scenario, we have constructed a function of speedE P E D C O P Y
against time, and represented our function with a table of values and a graph.
The previous example is based on a real world scenario and has limitations on the values of the x -variable. For example, a runner cannot have negative time (which would mean he is running backwards in time!), nor can he exceed the limits of human physiology (can a person run 100-meters in 5 seconds?).
However, we can apply the skills of constructing tables of values and plotting graphs to observe the behavior of rational functions.
Example 4.
Example 4. Represent the rational function given by
using a table of values and plot a graph of the function by connecting points.
Solution. Since we are now considering functions in general, we can find function values across more values of x . Let us construct a table of values for some x -values from -10 to 10:
0 2 4 6 8 10
1.22 1.29 1.4 1.67 3
0.33 0.6 0.71 0.78 0.82Plotting the points on a Cartesian plane we get:
D E P E D C O P Y
Connecting the points on this graph, we get:
Why would the graph unexpectedly break the smooth curve and jump from point E to point F? The answer is that it doesn’t! Let us take a look at the function again:
Observe that the function will be undefined at
. This means that there cannot be a line connecting point E and point F as this implies that there is a point in the graph of the function where
.We will cover this aspect of graphs of rational functions in a future lesson, so for now we just present a partial graph for the function above as follows:Example 5.
Example 5. Represent the rational function
using a table of values. Plot the points given in the table of values and sketch a graph by connecting the points.Solution. As we have seen in the previous example, we will need to take a look at the x -values which will make the denominator zero. In this function,
willmake the denominator zero. Taking function values for integers in
we get the following table of values:
2 3 4 5 6 7 8 9 10
0 6
0 1.33 2.57 3.75 4.89 6D E P E D C O P Y
Plotting the values above as points in the Cartesian plane:
We connect the dots to sketch the graph, but we keep in mind that
is notpart of the domain. For now we only connect those with values