Question

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

Answer 1

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]}