question about functions

Question

If f(4)=10, how do I write the function for it?

Answer 1

if it's a simple linear function through the origin then it's

y=ax

plug in the point (4,10)

10=a(4)

a = 10/4 = 5/2

so the function is

f(x) = 5x/2

but there are endless functions that could fit f(4) = 10

if it doesn't go through the origin and it's linear

more generally it would be

f(x) = 5x/2 + b where b is the y intercept, any real number

the graph is a straight line

It could be a quadratic function, a parabola, of 4 standard types, most common is y=ax^2

with vertex at the origin

then

10=a(4)^2 = 16a

a = 10/16 = 5/8

f(x) = 5x^2/8

but that's just the upward opening parabola with vertex at the origin

Aside from other parabolic functions, the function could be cubic or higher degree

f(x) = ax^3

10 = a(4^3) = 64a

a = 10/64 = 5/32

f(x) = 5x^3/32