If \(f(x)=\frac{16}{5+3x}\) , what is the value of \(\left[f^{-1}(2)\right]^{-2}\) ?
Let's first write this as
y = 16 / [ 5 + x ]
We need to find the inverse ....the idea is to isolate x and then "exchange" x and y
Multiply both sides by [ 5 + x ]
[ 5 + x ] y = 16 now....divide both sides by y
5 + x = 16/y subtract 5 from both sides
x = 16/y - 5 " exchange " x and y
y = 16/x - 5 get a common denominator on the right
y = 16/x - 5x / x
y = [ 16 - 5x ] / x and for y, write f-1(x)
f-1(x) = [ 16 - 5x ] / x
Now....we want to first find [ f-1(2 ) ]
This means that we want to put 2 into the inverse and evaluate it.....so we have
[ 16 - 5(2) ] / 2 = 6/2 = 3
So..... [ f-1(2 ) ] = 3
Now we want to find [ f-1(2 ) ] -2 which is just [ 3 ]-2
Remember that a-m = 1 / am
Therefore..... 3-2 = 1 / 32 = 1 / 9
So......[ f-1(2 ) ] -2 = 1 / 9
And that's it..!!!!