We have a Table here called Function A. So we need to know whether the table represents a linear or exponential relationship. Remember that:
A linear function has the form:
[tex]f(x)=mx+b[/tex] so if [tex]x[/tex] increases by a constant amount the functions value will have a common difference.
An exponential function has the form:
[tex]f(x)=ab^x[/tex] so if [tex]x[/tex] increases by a constant amount the functions value will have a common difference.
FUNCTION A)
EXPONENTIAL RELATIONSHIP.
Notice that in this case the function increases by a constant amount and in this case this amount is 3 because:
[tex]6-3=3 \\ \\ 9-6=3 \\ \\ 12-9=3[/tex]
As you can see, it is true that [tex]x[/tex] increase by a constant amount of 3. Hence, we need to analyze the function values.
First, let's see if the function values has a common difference:
[tex]9.252-6.083=3.169 \\ \\ 14.07-9.252=4.818 \\ \\ Stop \ here![/tex]
We have stopped because there is no any common difference. Let's try to see if there is common ratio:
[tex]\frac{9.252}{6.083}=1.52 \\ \\ \frac{14.07}{9.252}=1.52 \\ \\ \frac{21.4}{14.07}=1.52[/tex]
As you can see these function values have a common ratio. In conclusion THIS TABLE REPRESENTS AN EXPONENTIAL RELATIONSHIP.
FUNCTION B)
LINEAR RELATIONSHIP.
From the Table, it's easy to realize that this is a linear relationship by taking a look on the function values. Notice that [tex]x[/tex] increases here at a constant amount of 2 because:
[tex]6-4=2 \\ \\ 8-6=2 \\ \\ 10-8=2[/tex]
So, let's see if the function values has a common difference:
[tex]14-7=7 \\ \\ 21-14=7 \\ \\ 28-21=7[/tex]
As you can see these function values have a common difference. In conclusion THIS TABLE REPRESENTS A LINEAR RELATIONSHIP.
FUNCTION C)
EXPONENTIAL RELATIONSHIP.
It is likely that this is an exponential function, just take a look at the function values. Notice that [tex]x[/tex] increases here at a constant amount of 2 because:
[tex]2-0=2 \\ \\ 4-2=2 \\ \\ 6-2=2[/tex]
So, let's see if the function values has a common ratio:
[tex]\frac{4}{16}=0.25 \\ \\ \frac{1}{4}=0.25 \\ \\ \frac{0.25}{1}=0.25[/tex]
As you can see these function values have a common ratio. In conclusion THIS TABLE REPRESENTS AN EXPONENTIAL RELATIONSHIP.
