Solved Examples and Worksheet for Identifying Linear and Exponential Functions for the Given Data

Q1Choose the exponential model for the data shown.

A. f(x) = 2 · 3x
B. f(x) = 2 · 2x
C. f(x) = 2 · (12)x
D. f(x) = 2

Solutions
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Step: 1
Let the exponential function be f(x) = M (N)x.
Step: 2
At x = 0, f(x) = 2.
  [From the table.]
Step: 3
2 = M (N)0 ⇒ M = 2
  [Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 1, f(x) = 1.
  [From the table.]
Step: 5
1 = M (N)1
  [Substitute the values in the exponential function given in step 1.]
Step: 6
1 = 2N ⇒ N = 12
  [Substitute M = 2.]
Step: 7
Hence, the exponential function is f (x) = 2 · (12)x.
  [Substitute the values of M & N in the exponential function given in step 1.]
Correct Answer is :   f(x) = 2 · (12)x
Q2Which of the following exponential formulas represents the numerical data given?


A. g(x) = 2 (4)x
B. g(x) = 16(2)x
C. g(x) = 2 (14)x
D. g(x) = 14(2)x

Solutions
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Step: 1
Let the exponential function be g(x) = M (N)x.
Step: 2
At x = 0, g(x) = 14.
  [From the table.]
Step: 3
14 = M (N)0 ⇒ M = 14
  [Substitute the values in exponential function given in step 1.]
Step: 4
At x = 1, g(x) = 12.
  [From the table.]
Step: 5
12 = M (N)1
  [Substitute the values in exponential function given in step 1.]
Step: 6
= 14N
  [Substitute M = 14.]
Step: 7
⇒ N = 2
Step: 8
Hence, the exponential function is g(x) = 14 (2)x.
  [Substitute the values of M & N in the exponential function in step 1.]
Correct Answer is :   g(x) = 14(2)x
Q3Find the function that suits the given data.
(-2,3.6),(-1,2.6),(0,1.6),(1,0.6),(2,-0.4)

A. y = - x + 1
B. y = - x
C. y = - x + 1.6
D. y = - x + 1.5

Solutions
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Step: 1
Graph the line that best fit the data.
Step: 2
The two points that lie on the line are (0,1.6) and (1,0.6)
Step: 3
m = y2-y1x2-x1.
  [Slope formula.]
Step: 4
m = 0.6 - 1.61 - 0 = -11 = -1
  [Substitute and Simplify.]
Step: 5
Find the y intercept of the line using slope-intercept form.
Step: 6
y = mx + c
  [Write slope - intercept form.]
Step: 7
1.6 = (-1) (0) + c
  [Replace m with -1, x with 0 and y with 1.6.]
Step: 8
c = 1.6
Step: 9
The function that satisfies the given data is y = - x + 1.6.
Correct Answer is :   y = - x + 1.6
Q4The (input, output) pair shows the data of the volume of the air present inside a balloon as time elapses. Find the function for the data. Here input represents time in seconds and output represents volume in cubic m.
x , y = {(5, 13), (10,19) , (15, 127), (20, 181)}

A. y = (12) x5
B. y = (3) x5
C. y = (13) x5
D. y = (2) x5

Solutions
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Step: 1
The values obtained when x = 5,10, 15,20 in y = (13) x5 are 13 , 19 , 127, 181
Step: 2
The values obtained when x = 5, 10, 15, 20 in y = (12) x5 are 12 , 14 , 18, 116
Step: 3
The values obtained when x = 5, 10, 15, 20 in y = (2) x5 are 2,4,8,16
Step: 4
The values obtained when x =5,10,15,20 in y = (3) x5 are 3,9,27,81
Step: 5
The values of y obtained for the equation y = (13) x5 are same as that of the value given in the ordered pair.
Step: 6
So, the equation y = (13) x5 best represents the given set of ordered pair.
Correct Answer is :   y = (13) x5
Q5Which of the following function represents the data and find the value of f(6). (0,1) , (1, 32) (2 , 2) , (3, 52), (4, 3) , (5, 72)
A. y = x2 + 2,6
B. y = x2, 4
C. y = x + 1 , 2
D. y = x2 + 1 , 4

Solutions
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Step: 1
The values obtained when x = 0,1,2,3,4,5 in y = x2 + 1 are 1, 32, 2, 52 , 3, 72
Step: 2
The values obtained when x = 0, 1, 2, 3, 4, 5 in y =x + 1 are 1, 2 , 3 , 4 , 5 ,6
Step: 3
The values obtained when x = 0, 1, 2, 3, 4, 5 in y = x2 are 0, 12 , 1 , 32, 2, 52
Step: 4
The values obtained when x = 0, 1, 2, 3, 4, 5 in y = x2 + 2 are 2, 52 , 3, 72 , 4 , 92
Step: 5
The values of y obtained for the equation y = x2 + 1 are same as that of the values given in the ordered pair.
Step: 6
So, the equation y = x2 + 1 best represents the given set of ordered pair.
Step: 7
f(x) = x2 + 1
Step: 8
f(6) = 62 + 1 = 4
Correct Answer is :   y = x2 + 1 , 4
Q6Identify the function that satisfies the given (input, output) pairs and find the value of f(x) when x=5

A. f(x) = 0.5x+2, 4.5
B. f(x) = 0.5x+1, 1.5
C. f(x) = 5x+2, 3.5
D. f(x) = 0.5x, 2.5

Solutions
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Step: 1
The values obtained when x = -3,-2,-1,0,1,2,3 in y =0.5x +2 are -2.5,1,1.5,2,2.5,3,3.5
Step: 2
The values obtained when x=-3,-2,-1,0,1,2,3 in y=5x+2 are -13,-8,-3,2,7,12,17
Step: 3
The values obtained when x=-3,-2,-1,0,1,2,3 in y=0.5x+1 are -0.5,0,0.5,1.5,2,2.5
Step: 4
The values obtained when x=-3,-2,-1,0,1,2,3 in y=0.5 are -1.5,-1,-0.5,0,0.5,1,1.5
Step: 5
The values of y obtained for the equation y= 0.5x+2 are same as that of the values given in the ordered pair
Step: 6
So, the equation y=0.5x+2 best represents the given set of ordered pair
Step: 7
f(5)=0.5(5)+2 = 4.5
Correct Answer is :   f(x) = 0.5x+2, 4.5
Q7Construct an exponential function by given input-output pairs.
(-2,9),(-1,3),(0,1),(1,13),(2,19)

A. y = 3x
B. y= 2-x
C. y= 2x
D. y = 3-x

Solutions
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Step: 1
Let the exponential function be f(x) = M(N)x
Step: 2
At x = 0, f(x) = 1
  [From the ordered pair.]
Step: 3
1=M(N)0 → M = 1
  [Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 1, f(x) = 13
  [From the ordered pair.]
Step: 5
13 = 1(N)1 → N = 13
  [Substitute M=1.]
Step: 6
The exponential function y = 1(13)x = (3 -1)x = 3-x
Correct Answer is :   y = 3-x
Q8The pair of data shows the bus tickets sold in 4 months. Construct the function that matches the data and find how many tickets sold in 5 thmonth.(1,70),(2,35),(3,17.5),(4,8.75)
A. y = 120(12)t
B. y=140(12)t
C. y = 110(12) t
D. y=130(12)t

Solutions
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Step: 1
Let the exponential function be f(x) = M(N)t
Step: 2
At x=1, f(x)=70
  [From the ordered pair.]
Step: 3
70 = M(N)1 → M = 70N
  [Substitute the values in the exponential function given in step 1.]
Step: 4
At x = 2, f(x) = 35
  [From the ordered pair.]
Step: 5
35 = M(N)2
Step: 6
35 = 70N(N2)→ N = 12
  [Substitute M= 70N.]
Step: 7
M = 7012 = 140
  [Substitute N =12 .]
Step: 8
The function that matches the given data is y =140(12)t
Correct Answer is :   y=140(12)t
Q9Choose an exponential function that represents the given set of (input, output) pairs.
(Input, output) = {(-2, 11024),(-1, 1256),(0,164),(1,116),(2, 14)}

A. y = 4x - 1
B. y = 4 x - 2
C. y=4 x -3
D. y = 2x - 3

Solutions
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Step: 1
The values obtained when x = -2, -1 , 0, 1 ,2 in y=4 are 11024,1256,164, 116, 14
Step: 2
The values obtained when x= -2, -1, 0, 1, 2 in y=4x - 2 are 1256, 164,116,14,1,4
Step: 3
The values obtained when x= -2, -1, 0, 1, 2 in y=4x - 1 are 164, 116, 14, 1, 4
Step: 4
The values obtained when x= -2,-1,0,1,2 in y=2x-3 are 132, 116, 18, 14, 12 /32,1/16,1/8,1/4,1/2
Step: 5
The values of y obtained for the equation y=4x - 3 are same as that of the values given in the ordered pair.
Step: 6
The function y = 4x - 3 is best suits the given data.
Correct Answer is :   y=4 x -3
Q10The ordered pair shows the number of pupils in 4 classes. Find the function that best suits the given data.
(1,23),(2,43),(3,63),(4,83)

A. y=20x + 3
B. y = 10x + 2
C. y=10x + 3
D. y=20x+2

Solutions
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Step: 1
The values obtained when x=1,2,3,4 in y=20x + 3 are 23,43,63,83
Step: 2
The values obtained when x=1,2,3,4 in y=10x+3 are 13,23,33,43
Step: 3
The values obtained when x=1,2,3,4 in y=20x+2 are 22,42,62,82
Step: 4
The values obtained when x=1,2,3,4 in y=10x+2 are 12,22,32,42
Step: 5
The values of y obtained for the equation y=20x+3 are same as that of the values given in the ordered pair.
Step: 6
The function y=20x+3 is satisfies the given input-output pairs.
Correct Answer is :   y=20x + 3

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