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

Answer:

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

Step-by-step 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

$f^{-1}(x) = \frac{3^{x+6}-5}{4}$

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.

$f^{-1}(-4)=\frac{3^{-4+6}-5}{4}=\frac{3^2-5}{4}=\frac{9-5}{4}=\frac{4}{4} =1$

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.

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