Honors Math 2
Name:
Date:
Study Guide
Test: 4A (Lessons 4.0-4.05) and 4B (Lessons 4.06-4.10)
Topics covered:
 Find closed-form and recursive functions to fit input-output tables
 Use difference tables to determine whether a linear or quadratic function will fit a given table
 Use the up-and-over and hockey stick properties of difference tables
 Horizontal and vertical line tests
 Function notation
 Domain and range: algebraically, numerically, and graphically
 Inverse functions – Find the inverse of a function, graph the inverse of a function, determine whether the
inverse is a function
 Piecewise functions - write the equations, graph the functions, word problems
 New notation: composition symbol, inverse function notation
Tips for studying for your math test:
 Study by doing problems, not just reading over your notes.
 Look over your quizzes - make sure you know how to do those problems.
 Material you are responsible for includes: notes, handouts, and quizzes. Make sure you have looked over all
those items and that you know how to do all that work without the assistance of friends, tutors, notes, book,
etc. You won't have those resources for the test.
 Look in the book and at old handouts for more practice. There are plenty of problems that we have not yet
done! Some handouts are left unfinished, you have review problems on this handout, and there are many,
many problems in our book that we have not tried.
Review Questions - Show all work.
Please note: these problems are NOT intended to be inclusive of all the things you need to know for the
test – they are some practice problems to help you prepare.
1. Consider these function formulas:
f(x) = x  5
g(x) = x2 – 5
h(x) = (x – 5)2
Two of these could be formulas for functions that are inverses
of each other, if the domains are chosen appropriately.
Name the two functions and their domains, then sketch graphs
on the grid. Also sketch the reflection line relating the two
graphs as a dotted line.
2. Suppose j, k: R  R, j(x) = 2x + 1, and k(x) = 3 – x – x2
a. Find the ranges of j and k.
b. Find a formula for j  k (x) .
c. Find a formula for k  j (x) .
d. Find k(a + 1) – k(a).
e. Find j –1(25).
f. Let m = j 1  k  j. Find a formula for m(x).
3. Draw a set diagram (“potato-and-arrow diagram”) that represents the composition (g oh) ok for
functions g, h and k. Be sure to label your diagram appropriately.

4. Consider these three function graphs.
a. Only one of these function graphs could possibly be one-to-one. Which one?
b. Let f(x) be the function you chose in part a. On the same grid, draw a graph for f –1(x).
5. Fill in the following as they relate to f(x) graphed below:
6
4
2
-10
-5
5
-2
a. What is the domain?
b. What is the range?
c. f(-2) = _________________
d. f(7) = _________________
e. f(f(-2)) = ______________
f. f(_____) = -4
g. f(_____) = 0
h. Write the equation of f(x).
10
-4
-6
2
6. Consider the statement “The function f(x), has a domain of {1, 2, 3, 4, 5, 6, 7, 8, 9} and range of {1, 2,
3, 4, 5, 6, 7, 8, 9, 10}”
a. What is wrong with the statement?
b. How can you fix it?
7. What is the domain of each function?
a. p(x) 
1
5x 18
b. m(x) 
1
8. Suppose f ( x ) 
and g  x  
x 1

a. Find f(g(x)).
x . Find the following compositions and their domains.

b.
Find g(f(x)).
9. Values for h(x) and p (x ) are shown in the table below.
x
2
3
4
5
h(x)
3
5
7
9
p(x)
9
8
7
6
a.
b.
h(3)
5
x  9x 10
2
c.
p(h(3))
6
9
4
7
7
2
8
5
4
d.
h( p (h(5)))
9
3
6
p ( p ( p ( p (4))))
20
18
10. Graph this piecewise function:
16

2s 1
s  2


p(s)  x  1 2  s  3
1
x 1
s3

2
14
12
10
8
6
4

2
11. Find the formula for f
1
5
10
15
20
( x ) and state the domain of the function and its inverse.
b. f ( x) 
a. f ( x)  3 x 3  5
x3
x2
12. Use composition to confirm that f (x) and g (x) are inverses.
a. f ( x) 
x3
4
g ( x)  4 x  3
b. f ( x) 
x 1
x
g ( x) 
1
x 1
3
13. Given:
f ( x)  2 x
a. Write the function as a piecewise function. Is the inverse a function?
b. Sketch the inverse of the function.
c. State a domain restriction such that f (x) is one-to-one.
14. On a certain day, one British pound was worth 1.6603 US dollars. On that same day, one US dollar
was worth 1.5325 Canadian dollars. Write the rule for the following functions.
a. B, representing a conversion from x British pounds to US dollars
b. C, representing a conversion from x US dollars to Canadian dollars.
c. C oBx
d. What does C oBx represent?
k ( x)  2( x  4) 3 as the composition of the three functions below.
 15. Express the function
f ( x)  x 3 g ( x)  x  4

16. Write a closed-form and a recursive rule to
match the table below. Use either rule to find
a(15).
n
0
a(n)
1
1
5
2
25
3
125
4
625
5
3125
j ( x)  2 x
17. Make a difference table for the input-output table
below. Include enough columns to show a constant
difference. Use your difference table to find a closedform function that matches the table. Explain how you
used the information in the table to find the function.
x
0
c(x)
-3
1
4
2
15
3
30
4
49
18. Copy and complete the difference table below, then find the function that agrees with the table.
x
b(x)

0
1
2
1
2
3

2
8
3
3
4
5
41
More review (optional): “Mathematical Reflections” pp. 310–311.
4