How do you find the inverse of #f(x) = 200*3^(x/4) #?

1 Answer
Apr 7, 2018

See below

Explanation:

Given: #y= 200*3^(x/4)#

  1. Switch the x and the y:
    #x= 200*3^(y/4)#
  2. Solve for y by dividing by 200:
    #x/200=3^(y/4)#
  3. Apply logarithm:
    #log_3(x/200)= y/4#
  4. Multiply by 4 on both sides:
    #4log_3(x/200)= y#
  5. Write with inverse notation:
    #f^(-1)(x)=4log_3(x/200) #

  6. Symmetric across #y=x# check:
    graph{200*3^(x/4) [-71.4, 102.55, -11.7, 75.25]} graph{y=x [-62.74, 111.2, -64.9, 22.06]}
    graph{4((log(x/200))/log3) [-62.74, 111.2, -64.9, 22.06]}