Question

Larry is using an online calculator to calculate the outputs f(n) for different inputs n. The ordered pairs below show Larry's inputs and the corresponding outputs displayed by the calculator:

(1, 5), (2, 9), (3, 13), (4, 17)

Which of the following functions best represents the rule that the calculator uses to display the outputs?

a
f(n) = 5n − 1

b
f(n) = 5n + 1

c
f(n) = 4n + 1

d
f(n) = 4n − 1

Answer 1

Option c:

`f(n)=4n+1`

Explanation:

The functional relationship between two variables can be easily found if it's represented as a line.

Larry's online calculator collects these points

(1, 5), (2, 9), (3, 13), (4, 17)

We can see there is a linear relation because every time the first component increases by 1, the second increases by 4.

The equation of a line is given by

`f(n)=m.n+b`

Where m is the slope of the line and can be computed as

`\displaystyle m=(d-b)/(c-a)`

Where (a,b), (c,d) are two known points of the line. Let's use the first two points (1, 5), (2, 9)

`\displaystyle m=(9-5)/(2-1)=4`

We now know that

`f(n)=4n+b`

To compute the value of b, we use one of the points again, for example (1,5):

`5=4(1)+b => b=1`

The relation is

`f(n)=4n+1`

We can test our results by using other points like (3,13)

`f(3)=4(3)+1=13`

And also

`f(4)=4(4)+1=17`

All points belong to the same function or rule

`f(n)=4n+1`