The values of a function f(x) are given in the table below:
x=1, f(x)=3
x=2, f(x)=13
x=3. f(x)=8
x=5,f(x)=1
x=8,f(x)=0
x=13, f(x)=5
If f^-1 exists, what is
f^-1((f^-1(5)+f^-1(13))/ f^-1(1))?
I am not quite sure how to do this.
just the way you would evaluate any complicated expression. Start deep and work outward:
f^-1(5) = 13
f^-1(13) = 2
f^-1(1) = 5
So, you have
f^-1((13+2)/5)
=f^-1(3)
= 1
To find the answer to this question, we need to first determine the inverse function of f(x).
The inverse of a function switches the roles of x and f(x), so if we have f(x), the inverse function is denoted as f^(-1)(x).
To find f^(-1)(x), we need to switch the x-values and f(x)-values in the table given.
Original table:
x=1, f(x)=3
x=2, f(x)=13
x=3, f(x)=8
x=5, f(x)=1
x=8, f(x)=0
x=13, f(x)=5
New table after switching x and f(x):
x=3, f^(-1)(x)=1
x=13, f^(-1)(x)=2
x=8, f^(-1)(x)=3
x=1, f^(-1)(x)=5
x=0, f^(-1)(x)=8
x=5, f^(-1)(x)=13
Now let's solve the given expression: f^(-1)((f^(-1)(5) + f^(-1)(13)) / f^(-1)(1))
Step 1: Calculate f^(-1)(5) and f^(-1)(13)
From the new table above, f^(-1)(5) = 13 and f^(-1)(13) = 2
Step 2: Calculate f^(-1)(1)
From the new table above, f^(-1)(1) = 5
Step 3: Substitute the values into the expression
f^(-1)((f^(-1)(5) + f^(-1)(13)) / f^(-1)(1))
= f^(-1)((13 + 2) / 5)
= f^(-1)(15 / 5)
= f^(-1)(3)
Therefore, f^(-1)((f^(-1)(5) + f^(-1)(13)) / f^(-1)(1)) is equal to f^(-1)(3).