Determine whether a relation is a function. If the relation is a function, state its domain and range.|MATH EXAM
MATH 150 W01 Name ——————————————- Exam I
Fall, 2018 Dr. Mahmoud
1. Simplify the following expressions.
i. ( )
2 33 8
5
3x y x
x y
−
−
=
ii.
3 1
2 3
2
3 4
4 3
x x y
y
−
−
=
iii. 16 1464x y =
iv. ( )
6 2
2 2
x y
x =
+
v. 3 4− − − =
vi. ( ) 2
3 1 5 (1 )x x x− − − =
vii. ( )( ) =3 5
2524 yyy
viii. ( ) 2
2 3 5x− =
2. Perform the following operations:
i. = ++
− −
−
34
3
9
1 22 xxx
ii. = −
− −−
−
16
3
82
6 22 xxx
x
iii. = −
− 551 2
2
r
r
r
r
iv. =− +
−
4
1
2
1
x
x
3. Find the equation for the line with the given properties. Express your answer in point-slope form of the equation of a line.
(i) Graphed below.
(ii) Perpendicular to the line 12 += xy containing the point )3,1( − .
-3 -2 -1 1 2 3
-10
-5
5
10
15
20
4. Determine whether a relation is a function. If the relation is a function, state its domain and range.
5. This exercise assesses your understanding of the definition of a function. i. Sketch a curve in the coordinate plane that is not the graph of a function and justify your
answer.
ii. Give an example of a function whose domain consists of 5 numbers and whose range consists of 3 numbers.
iii. Give an example of a function whose domain is the set of all real numbers and whose range is ),1[ .
6. Find the domain of the function defined by the following equation:
(d) 3)( xxf −=
7. For the functions ( ) 2 4f x x= − and ( ) 3 6g x x= − + , find the following and also find the
domain:
8.
9. Which of the following are graphs of functions? In either case, state the domain and range of the relation. Using the definition of a function, state why the vertical – line test
works.
(a) (b) (c) (d)
10. Given the graph of the function, find each of the following:
The zeros of a function: If f (r ) = 0 for a real number r, then r is called __a zero____ of f.
11. For the function g below find the following:
12.
Definition: A function is even if, for every number x in its domain, the number – x is also
in the domain and f (−x) = f (x). So, in an even function, for every point (x, y) on the graph, the
point (-x, y) is also on the graph.
Theorem: A function is even if and only if its graph is __symmetric___ with respect to the
y- axis
Definition: A function is odd if, for every number x in its domain, the number – x is also
in the domain and f (−x) = − f (x). So, in an odd function, for every point (x, y) on the graph, the
point (-x, -y) is also on the graph.
Theorem: A function is odd if and only if its graph is symmetric with respect to the origin
13. Use the definitions of even and odd functions to determine algebraically whether each of the following functions is even, odd, or neither. Then graph to see symmetry.
Definitions: A function f is increasing on an open interval I if, for any choice of 1x and 2x
in I, with 1x < 2x we have 1 2( ) ( )f x f x . In other words, if a function is increasing, then, as the
values of x get bigger, the values of the function also get bigger.
A function f is decreasing on an open interval I if, for any choice of 1x and 2x in I, with
1x < 2x we have 1 2( ) ( )f x f x . In other words, if a function is decreasing, then, as the values of
x get bigger, the values of the function get smaller.
A function f is constant on an open interval I if, for all choices of x in I, the values of f(x)
Are stay the same.
14. Determine where the function is increasing, decreasing, or constant from its graph.
(a) Where is the function increasing?
(b) Where is the function decreasing?
(c) Where is the function constant?
15. For each of the functions in our library, fill in the table of values, graph the function, then complete the table of properties.
properties y x= 2y x= 3y x= y x= 3y x= 1y x
= y x=
Domain:
Range:
x-Intercepts
y- Intercepts
Symmetry: Interval the
function is
decreasing:
Interval the
function is
increasing:
f(x) y x= 2y x= 3y x= y x=
Graph
3y x= 1y
x =
y x=
Graph Piecewise – Defined Functions
16. The function f is defined as
(a) Find f (−2), f (1), and f (2)
(b) Determine the domain of f.
(c) Locate any intercepts
(d) Graph f.
Function Transformations
17. Using the graph of a parent function and the appropriate transformations, draw the graph of the following function. Make sure you draw all intermediate steps, indicate which
transformation was applied and label the coordinates of a relevant point.
21)( 3 −+−= xxf
18.
19. For each of the following quadratic equations determine their number of solutions (using the discriminant ) and where appropriate find the solution(s):
i. 0232 2 =−+ xx
ii. 012 =−+ xx
Inequalities Involving Quadratic Functions
20. Solve the inequality x2 + 3x − 28 0 using both methods above.
