Question

Determine the inverse of the function f(x) = log3(4x + 5) − 6.

f inverse of x is equal to 3 to the power of the quantity x minus 6 end quantity plus 5 all over 4
f inverse of x is equal to 3 to the power of the quantity x plus 6 end quantity minus 5 all over 4
f inverse of x is equal to 3 to the power of x plus 4 all over 4
f inverse of x is equal to the quantity x plus 6 end quantity cubed minus 5 all over 3

Answer 1

B-

Explanation:

I took the test

Answer 2

f inverse of x is equal to 3 to the power of the quantity x plus 6 end quantity minus 5 all over 4

Explanation:

The function
`f(x)=\log_3 (4x+5)-6` can be thought of as a series of steps, one operation at a time:

Start with x

Multiply by 4

Add 5

Take the
`\log_3`

Subtract 6

That gives you a function value
`f(x)`.

To get the inverse function, read that list from the bottom up (in reverse order, using inverse operations at each step).

Start with x (a bit confusing, because this x represents the function value you get at the end of the above list).

Add 6 (add is the inverse operation of subtract)

Raise 3 to the ...
`3^{\text{result of previous step}`

Subtract 5

Divide by 4

`f^(-1)(x) = (3^(x+6)-5)/(4)`

Let's test this out. Find f(1).

`f(1)=\log_3(4 \cdot 1 + 5)-6 =\log_3(9)-6=2-6=-4`

Now put -4 into the inverse function.

`f^(-1)(-4)=(3^(-4+6)-5)/(4)=(3^2-5)/(4)=(9-5)/(4)=(4)/(4) =1`

The final result is the number we started with when we put 1 into f(x).

Finding an inverse is reversing the action of the function f by doing inverse operations in "backwards" order.

A lot of authors have you do this by switching x and y in the formula for a function, then solving for x.

`y = \log_3(4x+5)-6\text{ switch x and y}\\\\x = \log_3(4y+5)-6\\\\x+6 = \log_3(4y+5)\\\\3^(x+6)=4y+5\\\\3^(x+6)-5=4y\\\\(3^(x+6)-5)/(4)=y`