Question

The spending limit on John’s credit card is given by the function f(x)=15,000+1.5x , where x is his monthly income. f^-1x . The variable x represents in the inverse function. If John's spending limit is $60,000, his monthly income is .

Answer 1

1.
"The spending limit on John’s credit card is given by the function f(x)=15,000+1.5x"

means that if the monthly income of John is $ 5,000 ,he can spend at most

f(5,000)=15,000+1.5*5,000=15,000+ 7,500=22, 500 (dollars)

Or for example

if Johns monthly income is $8,000, then he can spend at most

f(8,000)=15,000+1.5*8,000=15,000+ 12,000=27,000 (dollars)


2.
Now, assume that the maximum amount that John can spend is y.

Then, y=15,000+1.5x

we can express x, the monthly income, in terms of y by isolating x:

y=15,000+1.5x

1.5x = y-15,000



`x= (y-15,000)/(1.5)`

thus, in functional notation, x, the monthly income, is a function , say g, of variable y, the max amount:


`x=g(y)=(y-15,000)/(1.5)`

since we generally use the letter x for the variable of a function, we write g again as:


`g(x)=(x-15,000)/(1.5)`

for example :


`g(50,000)=(50,000-15,000)/(1.5)= (35,000)/(1.5)= 23,333`

tells us that if the maximum amount that John can spend is 50,000 $, then his monthly income is 23,333 $.

3.

If John's limit is $60,000, his monthly income is


`g(60,000)=(60,000-15,000)/(1.5)= (45,000)/(1.5)=30,000`
dollars.


Answer: $ 30,000

Remark: g is called the inverse function of f, since it undoes what f does.

instead of g(x), we could use the notation
`f ^(-1)(x)`