Question

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

Answer 1

Answer:

To determine the inverse of the given function, change f(x) to y, switch x and y and solve for y and f^-1(x)= ln(x+4)/2

Step-by-step explanation:

f(x) = e^2x - 4

to find the inverse let

y = e^2x - 4

Replace x and y

x = e^2y - 4

and now, solve to find value of y

x + 4 = e^2y

taking ln on both sides:

ln(x+4) = ln(e^2y)

ln(x+4) = 2y

=> y = ln(x+4)/2

=> f^-1(x)= ln(x+4)/2

So, To determine the inverse of the given function, change f(x) to y, switch x and y and solve for y and f^-1(x)= ln(x+4)/2

Answer 2

Answer with explanation:

The given function is:

    $\rightarrow y=e^{2x}-4\\\\\rightarrow y+4=e^{2x}\\\\\text{Taking log on both sides}\\\\\rightarrow \log(y+4)=2 x \log e\\\\\rightarrow \log(y+4)=2 x\\\\\rightarrow x=\frac{\log(y+4)}{2}$

To determine the inverse , change x to y and y to x of the above equation

      $\rightarrow y=\frac{\log(x+4)}{2}$

To determine the inverse of the given function change f(x) to y, switch  x

and y, and solve for y.

or

this is the way described above.

$\rightarrow y=e^{2x}-4\\\\\text{Switching x and y}\\\\\rightarrow x=e^{2y}-4\\\\\rightarrow x+4=e^{2y}\\\\\text{Taking log on both sides}\\\\\rightarrow \log(x+4)=2 y \log e\\\\\rightarrow \log(x+4)=2 y\\\\\rightarrow y=\frac{\log(x+4)}{2}$

The resulting function can be written as:

    $f^{-1}x=\frac{\log(x+4)}{2}$