F(x) is a linear function with f(2)=5 and f(4)=9. How do I solve f(10)?

Question

F(x) is a linear function with f(2)=5 and f(4)=9. How do I solve f(10)?

Answer 1

Another way is to find linear function using the point-slope form of a line.

 

f(x) - f₁(x) = m(x - x₁)

 

where:

m =  slope between the points

(x₁ , f₁(x)) is the any of the given points

 

Our two points are  (2, 5) and (4, 9).

 

m = (9 - 5) / (4 - 2)

m = 4 / 2

m = 2

 

 

Plugging this slope and the first point into the formula,

 

f(x) - 5 = 2(x - 2)

 

f(x) = 2(x - 2) + 5

 

f(x) = 2x - 4 + 5

 

f(x) = 2x + 1

 

 

Now evaluate the bolded f(x) when x=10.

Answer 2

To be a linear function, f(x) must have the form

 

f(x) = ax + b

 

Then f(2)=5 and f(4)=9 can be stated as

 

5 = 2a + b

9 = 4a + b

 

Subtracting the top equation from the bottom equation gives

 

4 = 2a

a = 2

 

Substituting this value of a in the top equation gives

 

5 = 2(2) + b

b = 1

 

Finally,

 

f(x) = 2x + 1

 

Then

 

f(10) = 2(10) + 1 = 21