Download Review: Functions from a variety of viewpoints

Review: Functions from a variety
of viewpoints
A real-valued function f of a real variable x is
a rule that assigns a single real number y = f(x)
to each real number x within a specified set of
numbers.
The variable x is called the independent
variable. The set of numbers that may be
specified for x is called the domain of f. A
function cannot be evaluated at a point outside its domain. We sometimes say the value
of a function at such a point is undefined.
When we assign a second variable to the value of the rule f(x), ie, y = f(x), we call y the
dependent variable. Do not become hooked on always using x and y for the variable
names, but know the difference between y vs. x and x vs. y.
A function can be specified:
Numerically - by means of a table
Algebraically - by means of a formula
Graphically - by means of a graph.
Note on Domains
The domain of a function is not always specified explicitly; if no domain is specified for
the function f, we take the domain to be the largest set of numbers x in which f(x) makes
sense. This "largest possible domain" is sometimes called the natural domain.
Exercises
1. Suppose that the function f is numerically specified by the following table. Nothing is
known about the values of the function at points other than the x values shown.
x
f(x)
0
1
2
3
4
3.01
-1.03
2.22
0.01
0.0
Then f(0) is the value of the function when x = 0. Complete the missing values in
the table below:
f(0)
Obtain these values directly from the
f(1)
function definition table, if possible
f(5)
f(2) + f(1)
Calculate these values
f(3) - 3
4 f(4)
2. Suppose that a different function f is algebraically specified by f(x) = 3x2  4x + 1.
Note: given a function that has values for all real numbers x, there are no
restrictions on the domain.
f(2) =
f(0) – f(1) =
f(1) =
f(4) – 2 =
3. The following table gives the per capita Gross Domestic Product (GDP) in $ US for
the Unites States since 1900.*
Year
t
0
1900 (1900)
=0
10
20
30
40
50
60
70
80
90
100
(2000)
GDP 6608.7 8019.0 8969.9 10036.7 11324.2 15445.1 18299.6 24279.9 30009.9 37478.9 46366.0
G(t)
a. What was the average change in GDP per year over the entire interval?
b. During which decade was the change in GDP the greatest?
c. During which decade was the change in GDP the smallest?
d. Qualitatively, how has the rate of change in GDP been varying as a function of
time?
e. Which of the following model functions for G(t) might best fits the given data?
(Use Mathematica or your calculator to test these functions). Note the ‘best fit’
function does not necessarily equal the given values at any of the given points.
A
B
Linear with positive slope
G(t) = a + bt
Quadratic
G(t) = a + bt + ct2
C
Logarithmic
G(t) = a + b ln kt
D
Exponential
G(t) = a e kt
f. Produce a plot showing your best fit function and the data points on the same set
of axes. Now use your function to predict GDP in the year 2010. 49258.7
4. Suppose that the function f is specified algebraically by the formula f(x) = x2/(x2 + 1)
with a domain of [-1, 10).
Note: The domain restriction in terms of brackets means that the square bracket
indicates that -1 is included in the domain, and the round bracket after the 10
indicates that 10 is not included.
Calculate the following values. If you use decimals, provide 3 places of accuracy.
Answer "undefined" if the function is not defined at the given value of x.
f(1) =
f(-2) =
f(2) =
f(.5) =
2 f(2) =
f(-1) + f(0) =
f(0) =
f(10) =
Piecewise Functions
5. Sometimes we need more than a single formula to specify a function algebraically:
US Federal Income Tax is calculated according to a system of income brackets. For
single tax payers in 2014, these brackets were:
10%
15%
25%
28%
33%
35%
39.6%
$1 - $9,075
$9,076-$36,900
$36,901 to $89,350
$89,351 to $186,350
$186,351 to $405,100
$405,101 to $406,750
over $406,750
where the percentage on the left is the tax rate on income earned within the boundaries of
the bracket (the range of incomes on the right). As an example, a person with taxable
income of $10000 paid 10% of the first $9075 plus 15% of the difference between
$10000 and $9075 for a total tax of $1046 (taxes are usually rounded to the nearest whole
dollar). We refer to the top tax rate (in this case 15%) as the marginal tax rate and the
average (in this case 10.46%) as the effective tax rate. Take-home pay is your taxable
income less any tax you have to pay.
a. Write a piecewise function that gives the tax to be paid on income up to and
including the top of the 25% bracket.
b. Suppose your taxable income is $88,000 and you are offered a 3% pay raise. By
the time you factor in the additional taxes, what is the actual % increase in your
take-home pay?
c. Proponents of the so-called ‘flat tax’ argue that a single tax rate would be a fairer
system, eliminating the distinction between marginal and effective rates. If the flat tax
rate is 25%, does a person with $50,000 in taxable income pay more or less tax under
this system? Does a person with $500,000 in taxable income pay more or less tax?
Defining a function
graphically
Pct GDP
US Military Spending as Pct of GDP
5
6. The graph at right shows the
percentage of GDP that is the total
military spending by the US.
Data source:
http://www.gapminder.org/data/
4
3
2
1
1990
1995
2000
2005
2010
a. Suppose we only considered the
two endpoints (1988, 5.71) and (2011, 4.71). What level of military spending would we
predict for the year 2020?
b. Suppose we added just the point for the year (2000, 3.02) to the endpoints. What type
of function could we use to make a prediction for 2020?
c. Suppose we decided that there was some periodic relationship to these data. What type
of function could we use? What would be the approximate period (years between peak
values)? Qualitatively, what would we predict for 2020?
7. Let f(x) = x + 1/x.
Here is a graph of a few points:
Clearly, the function has a domain restriction, in that the
independent variable x may not equal 0.
Here are some candidates for the graph of the function.
How can you decide which one is the correct graph?
year
Symmetry of functions

A function f is said to have even
symmetry if for all x in the domain
of f, f(-x) = f(x). Even symmetry
is recognized graphically by a
reflection in the y-axis.

A function f is said to have odd
symmetry if for all x in the domain
of f, f(-x) = -f(x). Odd symmetry is
recognized graphically be a
reflection in the origin.
8. Be able to determine both graphically and algebraically whether a function has even,
odd, or no symmetry. Make a list of examples of functions with odd symmetry and look
for a common characteristic. Do the same for functions with even symmetry.
a. Write an algebraic or logical statement or Mathematica function that will test
another function for odd/even symmetry without making a plot. Your statement
might look like “If ____ then ‘the function is odd’ or else ‘the function is even.’”
b. How would you test for symmetry in the
x-axis?
c. What is the symmetry of y = |x – 1| ?
Operations on functions
Given a function, we may create a new function by manipulating it. You should
understand what the parameters a , b and c do to the graph of any function f(x) in the
following:
a f(bx) + c
a f(x + c)
a f(bx - c)
9. Using the function y = x3 as a base, use Mathematica or Wolfram Alpha to explore
each of the above manipulations. Plot (on the same axes) the base function and a set of
transformations using sample values for a, b and c such as –1, ½, 1 and 2. This is also a
good time to learn how to label your axes and annotate your plots.
Composite functions: functions of function values
We often see one function written so that its domain is the range of another function.
This frequently takes the form y = f(g(x)).
In this notation, the function g is evaluated at the value x. Then the function f is
evaluated at the value g(x). We sometimes call g(x) the inner function and f(x) the outer
function. Any restrictions in the range of g(x) become domain restrictions on f(x).
10. Use this notation to write functional definitions for the following statements. In each
case, identify the inner function as g(x) and the outer function as f(x). Note any domain
restrictions.
a. Add 5 to the square of x.
b. Subtract 17 from the inverse of the sum of the square of x plus 1.
c. Take the cube root of the sum of the square of the quantity resulting from the sum
of x and 9.
d. Let f(x) = 2+x2 and g(x) = x2 – x – 1.
Evaluate f(g(2)), f(2), g(2), g(f(2)).
Transforms
An algebraic transformation is a manipulation of terms in a function that may lead to an
alteration in the shape of its graph. The result need not be a function.
A translation is a manipulation of terms in a function that moves it left, right, up or
down. For the function y = f(x), what do each of the following translations do?
y = f(x – h)
Moves the graph h units _______________
y = f(x + h)
Moves the graph h units _______________
y = f(x) + k
Moves the graph k units ___________
y = f(x) - k
Moves the graph k units ___________
Examples
1. Write the equation of the graph of
a) y = x2 after a horizontal translation of 3 units to the right
b) y = 10x after a vertical translation of 2 units up
c) y = x ½ after a horizontal translation of 4 units to the left and a vertical translation
of 3 units down
2. A train leaves Chicago, heading for Houston, at 2 pm. It travels with an average speed
of 30 km/hour for 2 ½ hours. At 2:45, a second train leaves Chicago, traveling with an
average speed of 40 kph for a period of 1 hour 15 minutes. Write an expression for the
distance traveled by each train over time period between 2 and 4 pm.
Reflections
An algebraic reflection flips a graph about an axis or line. Which line is the ‘mirror’ for
each of the following reflections?
y = f(x) and y = -f(x)
Are reflections about _________
y = f(x) and y = f(-x)
Are reflections about _________
y = f(x) and x = f(y)
Are reflections in the line _______
Any point which does not move after the reflection is called an invariant point. There
may or may not be any invariant points in a given transformation. For each of the above,
determine one invariant (if any exist).
Examples
1. Given y = f(x) = x3 – 8
a. write the equation of the reflection of y about the x axis
b. write the equation of the reflection of y about the y axis
2. a. For y = f(x) = x2, which reflection would result in the exact same graph?
b. For y = g(x) = x3, which reflection would result in the exact same graph?
3. The function y = h(x) is reflected as follows. Given that there are invariant points
under each reflection, describe the location of these points.
a. y = -f(x)
b. y = f(-x)
c. x = f(y)
4. Write the equation of the function y = |x + 1| after a reflection in the x axis.
5. Write the equation of the function y  x  2 after a reflection in the line y = x.
Stretch and squeeze
Multiplication of a function or a variable by a constant stretches or squeezes the graph of
the function. Describe each operation by the direction and whether it is a stretch or a
squeeze.
Each of the following operations transform y = f(x) to y = a f(x)
Is a vertical squeeze
0<a<1
a>1
Is a vertical stretch
a<0
Is a reflection with stretch or squeeze
Each of the following operations transform y = f(x) to y = f(bx)
Is a horizontal squeeze
0<b<1
b>1
Is a horizontal stretch
b<0
May be a reflection with stretch or squeeze (or
may not be defined!)
Note: some authors prefer to just use the term stretch. If so, a stretch by ½ is really a
squeeze, but a stretch by 3 is still a stretch. In the cases of the negative constant, the
stretch or squeeze is by a factor of the absolute value of the constant, with an appropriate
reflection.
Examples
1. What happens to the graph of y = f(x) if the following replacements are made?
a. replace x with x/2
b. replace y with 4y
c. replace y with -2y and x with 4x
d. replace y with y - 4 and x with –x/4
2. The graph of y = f(x) is squeezed vertically by a factor of 2 and stretched horizontally
by a factor of 3. If the final expression is y = a f(bx), what is the value of a – b ?
Graphically speaking
1. Given the graph as shown, representing a portion of the unit circle:
Sketch the result after a horizontal stretch by a factor of 3 and a vertical stretch by a
factor of 2. What is the equation of the stretched curve? Then consider the original
graph with these two stretches applied in the reverse order. Are the two altered graphs
identical?
2. Which combinations of transformations are independent of the order of application?
3. Describe the roles of the constants a, b, h and k in the following expression:
y=f(x) is transformed to y = a f[b(x – h)] + k
Reciprocals
1
. This is not the same
f ( x)
as y = f-1(x), which is the inverse function (the operations that “undo” the work of
function f).
For the function y = f(x), the reciprocal transformation is y 
1. Given y = x + 3, sketch the graph of the reciprocal of f(x). How is this transformation
differ from translation, stretch and squeeze and reflection? What happens to the
reciprocal where y = f(x) = 0? What happens where y = f(x) approaches positive or
negative infinity?
2. How are the domain and range of y = f(x) affected by the reciprocal operation?
Consider y = x2 and y = x3.
3. How would the reciprocal of an absolute value function, y = |x|, be formed?
Inversion
Given y = f(x), find the inverse function f-1(x) by the following procedure:
1. Swap occurrences of y and x.
2. Solve the resulting expression for y.
3. Note any restrictions on domain.
Examples
Find the inverse of a. y = x3 + 5
b. y = 3x - 4
c. y = a (bx) ½