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


$x= \frac{y-15,000}{1.5}$

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

$x=g(y)=\frac{y-15,000}{1.5}$

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

$g(x)=\frac{x-15,000}{1.5}$

for example :

$g(50,000)=\frac{50,000-15,000}{1.5}= \frac{35,000}{1.5}= 23,333$

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 

$g(60,000)=\frac{60,000-15,000}{1.5}= \frac{45,000}{1.5}=30,000$
 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 $f ^{-1}(x)$