Determine which function(s) are exponential. Select all that apply.

Answer:

option B and option C represent an exponential function.

Step-by-step explanation:

We know that following are some of the properties of different functions.

Linear Function:

The constant change in x and y

Quadratic Function:

The constant change in x

The constant 2nd differences in y

Exponential Function:

The constant change in x

The constant ratio in y

Determining the nature of the exponential function

Option A)

x       0          1          2         3           4

y       0          1           4         9          16

Here:

There is a constant change in x:

1 - 0 = 1;  2 - 1 = 1;  3 - 2 = 1;  4 - 3 = 1

BUT, there is NO CONSTANT ratio, as

1/0 = ∞,  4/1 = 4,  9/4 ; 16/9

Thus,

Table A DOES NOT represents an exponential function.

Option B)

x       0          1          2         3           4

y       1/3        1           3        9           27

Here:

There is a constant change in x:

1 - 0 = 1;  2 - 1 = 1;  3 - 2 = 1;  4 - 3 = 1

There is also a constant ratio, as

1/[1/3] = 3,  3/1 = 3,  9/3 = 3; 27/9 = 3

As there is constant change in x as well a constant ratio.

Thus,

Table B REPRESENTS an exponential function.

Option C)

x       0          1          2         3           4

y       5          5/2      5/4      5/8       5/16

Here:

There is a constant change in x:

1 - 0 = 1;  2 - 1 = 1;  3 - 2 = 1;  4 - 3 = 1

There is also a constant ratio, as

[5/2]/5= 1/2,

[5/4]/[5/2] = 1/2

[5/8]/[5/4] = 1/2

[5/16]/[5/8] = 1/2

As there is constant change in x as well a constant ratio.

Thus,

Table C REPRESENTS an exponential function.

Option D)

x       0          1          2         3           4

y       4.5       4        3.5       3          2.5

Here:

There is a constant change in x:

1 - 0 = 1;  2 - 1 = 1;  3 - 2 = 1;  4 - 3 = 1

BUT, there is NO CONSTANT ratio, as

4/4.5 = 4/[45/10] = 40/45 = 8/9

3.5/4 = [35/10]/4 = 35/40 = 7/8

3/3.5 = 3/[35/10] = 30/35 = 6/7

2.5/3 = [25/10]/3 = 25/30 = 5/6

Table D  DOES NOT represent an exponential function.

Conclusion:

Therefore, option B and option C represent an exponential function.

The functions that are exponential would be:

B). [tex]x[/tex]   0          1          2         3           4

   [tex]y[/tex]     1/3        1           3        9          27

C). [tex]x[/tex]       0          1          2         3           4

      [tex]y[/tex]     5          5/2      5/4      5/8       5/16

What are Exponential Functions?

B). Given that,

[tex]x[/tex]   0          1          2         3           4

 [tex]y[/tex]     1/3        1           3        9          27

As we see,

The value of [tex]x[/tex] undergoes a constant change of 1

[tex]1 - 0 = 1; \\2 - 1 = 1; \\3 - 2 = 1; \\4 - 3 = 1[/tex]

The ratio also remains constant

[tex]1/[1/3] = 3, \\3/1 = 3, \\9/3 = 3; \\27/9 = 3[/tex]

Thus, it exemplifies an exponential function.

Similarly,

C). [tex]x[/tex]       0          1          2         3           4

      [tex]y[/tex]     5          5/2      5/4      5/8       5/16

As we see,

The value of [tex]x[/tex] undergoes a constant change of 1

[tex]1 - 0 = 1; \\2 - 1 = 1; \\3 - 2 = 1; \\4 - 3 = 1[/tex]

The ratio also remains constant

[tex][5/2]/5= 1/2\\\\[5/4]/[5/2] = 1/2\\\\[5/8]/[5/4] = 1/2\\\\[5/16]/[5/8] = 1/2[/tex]

Thus, it exemplifies an exponential function.

Thus, options B and C are the correct answers as the other two have a constant change but not a constant ratio.

Learn more about "exponential function" here:

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