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Linear Functions
Linear Functions: Problems with Solutions
Problem 1
The proportional relation between distance traveled and the amount of time is shown in the following picture.
Which of the statements is true?
(A) The y coordinate of point A represents the distance traveled in 4 hours.
(B) The distance traveled in 1 hour is 60 kilometres.
(C) It is impossible to determine the distance if we have traveled zero hours.
(D) None of these is correct.
Solution:
Answer: (D) No answer is correct.
Point A has coordinates $(5, 400)$. It means that for 5 hours we travel 400 km.
Problem 2
The graph visualize an emplolyee pay based on his working hours.
Determine the rate per hour.
Solution:
The $x$ axis shows hours and $y$ axis the pay.
The point $(1,40)$ shows that if he works one hour he will be paid \$40.
The point $(2,80)$ shows that if he works 2 hours he will be paid \$80.
To know the payments per hour, we must measure the vertical distance between the points $(1,40)$ and $(2,80)$.
Distance $=y_{2}-y_{1} = 80-40$
The payment is 40 dollars per hour.
Problem 3
The following graph shows the relation between the distance traveled by a taxi and the total cost of the service.
Which of the following about the point A is true?
(A) A taxi service of 8 km has a cost of \$5
(B) A taxi service of 20 km has a cost of \$8
(C) A taxi service of 8 km has a cost of \$20
(D) None is correct.
Solution:
Correct: (B) A taxi service of 20 km has a cost of \$8.
The point A $(20, 8)$ shows that the cost of the taxi service of 20 kilometers is 8 dollars.
Problem 4
The following table represents the relation between x, y.
May this table represent a function?
X
Y
5
8
10
13
15
18
20
23
21
24
25
28
Yes
No
Solution:
The definition of a function is the following:
For each point of the starting set corresponds exactly one value of the arrival set. In the table this condition is met, for every value of x there are no two different values of y, therefore this table represents a function that relates the variable x, y.
Problem 5
Which of these relations is a function?
A
B
-1
0
-1
1
0
3
0
2
1
4
2
2
A
B
-1
0
-1
0
0
3
0
3
1
4
2
2
X
Y
5
10
15
15
20
25
25
35
25
24
25
28
(A) TABLE I and TABLE II
(B) TABLE I and TABLE III
(C) TABLE II and TABLE III
(D) None of these is correct.
Solution:
Answer: (B) TABLE I and TABLE III
The definition of a function is the following:
for every point of the first set corresponds a single value of the second set.
In table 1 the points $(-1,0)$ and $(-1,1)$ do not satisfy this condition, because when $x=-1$ there are two different values of $y$, also in table III, the points $(25,24)$ and $(25,28)$ do not meet the definition of a function.
Problem 6
Which of these relations is a function?
X
Y
5
5
10
10
15
15
20
20
21
21
25
25
X
Y
5
5
10
5
15
5
20
5
21
5
25
5
(A) TABLE I and TABLE II
(B) TABLE I
(C) TABLE II
(D) None of them.
Solution:
Both tables represent functions. In the first case we see the function identity to each value of x, the same value of y corresponds to it, this function is called identity function. In the second table we have the constant function, to all the values of x corresponds the same value of y, in both cases we have a function.
Problem 7
Let A and B be sets with elements:
$A=\left\{ 1,3,5,6,8\right\}$ $B=\left\{ a,b,d,f\right\} $
Is the set of ordered pairs a function or a relation between the sets A, B?
$R=\left\{ (1,b);(3,d);(6,f)\right\} $
Yes
No
Solution:
For each element of $x$ corresponds a single element of $y$, for this reason the set of ordered pairs represents a function from set $A$ to set $B$
Problem 8
Does the point $(1,3)$ belong to the graph of the function $y=2x-1$ ?
Yes
No
Solution:
In the linear function $y=2x-1$ we must substitute the value of $x$ by 1. $y=2(1)-1\Longrightarrow y=1$
Note that the graph of the function passes through the point $(1,1)$, but does not go through the point $(1,3$. Then we can conclude that the point $(1,3)$ does not belongs to the graph of the function $y=2x-1$
Problem 9
Which of the points $(1,3)$, $(0,3)$, $(2,0)$, $(0,2)$, do not belong to the graph of the function $y=x+2$?
(A) $(1,3)$ and $(0,3)$
(B) $(1,3)$ and $(0,2)$
(C) $(0,3)$ and $(2,0)$
(D) all points belong to the graph
Solution:
The correct answer is (B) $(1,3)$ and $(0,2)$.
$y=(1)+2 = 3 $ and
$y=(0)+2 =2$
So the the points $(1,3)$ and $(0,2)$ belongs to the graph of the function $y=x+1$.
You can check that none of the other points belong to the graph.
Problem 10
Given the function $y=3x-\frac{2}{5}$. Evaluate the function at x = 5.
(A) $\frac{5}{3}$
(B) $\frac{6}{5}$
(C) $\frac{3}{2}$
(D) $\frac{73}{5}$
Solution:
We must replace x by 5 in the definition of the function $y=3x-\frac{2}{5}$
$y=3(5)-\frac{2}{5}=15-\frac{2}{5}=\frac{75-2}{5}=\frac{73}{5}$
Problem 11
If $y=\frac{2x-1}{3}$, what is the value of $y$ when $x=1$?
$\frac{2}{3}$
$1$
$\frac{1}{3}$
$\frac{4}{3}$
Solution:
We must replace x by 1
$y=\frac{2(1)-1}{3}=\frac{1}{3}$
Problem 12
Which of following ordered pairs satisfy the function $y=5x-2$ ?
(A) $(0,-2)$ and $(1,3)$
(B) $(1,3)$ and $(0,2)$
(C) $(5,2)$ and $(2,5)$
(D) None of them
Solution:
Correct answer: (A) $(0,-2)$ and $(1,3)$
We must check if in the function $y=5x-2$ we replace x by 0 we get $y=-2$ and
if we replace x by 1 we get $y=3$.
For x = 0 we get
$y=5(0)-2 = 0-2 =-2$. So the the point $(0,-2)$ satisfies the function $y=5x-2$
and also $y=5(1)-2=5-2=3$ which shows that the point $(1,3)$ satisfies the function.
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