Relations and Functions Worksheet

Relations and Functions Worksheet
• Page 1
1.
A relation is observed between the number of days of preparation by a student for math, science and statistics tests and the points he scored. The relation is: {(3, 70), (4, 90), (2, 52)}. Determine if this relation is a function or not.
 a. yes b. no

Solution:

The given ordered pairs are: {(3, 70), (4, 90), (2, 52)}

A relation is a function if each input corresponds to one and only one output.

So, the given relation is a function.

2.
Determine whether the relation {($x$, $y$): $x$ = 10} is a function.
 a. yes b. no

Solution:

{(x, y): x = 10}

{(x, y): x = 10} is the set of all ordered pairs whose x-coordinate is 10 for any value of y.

That is, {_ _ _ _ (10, - 1), (10, 0), (10, 1), (10, 2), (10, 3) _ _ _ _ _ _}.

A relation is a function if each value of x corresponds to one and only one value of y.

So, the given relation is not a function.

3.
Find the domain and range of the relation graphed.

 a. domain = {3, 0, 4, 2}; range = {0, 1, 2, - 2, - 3} b. domain = {0, 1, 2, - 2, - 3}; range = {3, 0, 4, 2} c. domain = {2, 4}; range = {- 3, 2} d. domain = {1,- 2}; range = {3}

Solution:

The ordered pairs from the given graph are {(0, 3), (1, 0), (2, 4), (- 2, 0), (- 3, 2)}

The domain of a relation is the set of all the first coordinates of the ordered pairs. The range of a relation is the set of all the second coordinates of the ordered pairs.

The domain is {0, 1, 2, - 2, - 3}.

The range is {3, 0, 4, 2}.

4.
Determine whether the relation graphed is a function or not.

 a. yes b. no

Solution:

A relation is a function if each value of x corresponds to one and only one value of y.

But in the graph, x = 2 corresponds to more than one y- value.

So, it is not a function.

5.
Draw the mapping diagram for the relation and determine whether it is a function or not.
{(2, 4), (- 8, 0), (1, 5), (3, 1)}

 a. Figure 1, Relation is a function b. Figure 3, Relation is a function c. Figure 4, Relation is not a function d. Figure 2, Relation is not a function

Solution:

Draw the mapping diagram for the given relation.

A relation is a function if each element in the domain is paired with one and only one element in the range.

From the mapping diagram, it can be observed that the given relation is a function.

6.
In the relation {($x$, $y$): $x$ = 12$y$²}, is $x$ a function of $y$?
 a. No b. Yes

Solution:

{(x, y): x = 12y²}

A relation is a function if each value of x corresponds to one and only one value of y.

But here for two outputs, we have the same input.

For example: For x = 48, we have y = 2 and y = - 2

So, it is not a function.

7.
Identify a rule for the relation {(- 3, 33), (- 2, 22), (0, 0), (2, 22), (3, 33)}.
 a. {($x$, $y$): $y$ = - | 11$x$ |, $x$ = - 3, - 2, 0, 2, 3} b. {($x$, $y$): $y$ = - 11$x$, $x$ = - 3, - 2, 0, 2, 3} c. {($x$, $y$): $y$ = 11$x$, $x$ = - 3, - 2, 0, 2, 3} d. {($x$, $y$): $y$ = | 11$x$ |, $x$ = - 3, - 2, 0, 2, 3}

Solution:

{(- 3, 33), (- 2, 22), (0, 0), (2, 22), (3, 33)}

From the given relation, it can be observed that the y - coordinates in each ordered pair are absolute value of 11 times the x - coordinates.

So, the relation is: {(x, y): y = | 11x |, x = - 3, - 2, 0, 2, 3}.

8.
Laura is building rectangular boxes of different sizes. The length of the box '$y$' is 8 times its width '$x$'. If the widths she is using are 5 cm, 7 cm and 9 cm, then choose a rule for the relation between the length and the width and determine if this relation is a function.
 a. {($x$, $y$): $x$ = 8$y$, $x$ = 5, 7, 9}, not a function b. {($x$, $y$): $x$ = 8$y$, $x$ = 5, 7, 9}, function c. {($x$, $y$): $y$ = 8$x$, $x$ = 5, 7, 9}, function d. {($x$, $y$): $y$ = 8$x$, $x$ = 5, 7, 9}, not a function

Solution:

Length of the box = y

Width of the box = x

y = 8x

The relation is: {(x, y): y = 8x, x = 5, 7, 9}

The Ã¢â‚¬ËœyÃ¢â‚¬â„¢ values are: y = 40, 56, 72

The ordered pairs are (5, 40), (7, 56), (9, 72)

A relation is a function if each value of x corresponds to one and only one value of y.

So, the given relation is a function.

9.
Evaluate 6$f$(10), if $f$($x$) = 10$x$² + 6$x$ - 4.
 a. 6336 b. 1056 c. 10640 d. 1064

Solution:

f (x) = 10x² + 6x - 4

f (10) = 10(10)² + 6(10) - 4 = 1000 + 60 - 4
[Replace x with 10.]

f (10) = 1056
[Simplify.]

Therefore, 6f (10) = 6(1056) = 6336
[Simplify.]

10.
Evaluate $g$(3$x$) + $h$(- 10), if $g$($x$) = - 8$x$ + 16 and $h$($x$)= 5$x$² + 2$x$ + 8.
 a. - 24$x$ - 504 b. - 24$x$ + 504 c. 24$x$ - 504 d. 24$x$ + 504

Solution:

g(x) = - 8x + 16

g(3x) = - 8(3x) + 16
[Replace x with 3x in g(x).]

= - 24x + 16

h(x) = 5x² + 2x + 8

h (- 10) = 5(- 10)² + 2(- 10) + 8
[Replace x with - 10 in h(x).]

= 500 - 20 + 8

h (- 10) = 488
[simplify.]

g(3x) + h (- 10) = - 24x + 16 + 488 = - 24x + 504