Question

Which of the following equations represents the inverse of y = e2x - 9?

Answer 1

Answer:

$\frac{1}{2}ln(x+9) = y$

Step-by-step explanation:

We have given an equation.

$y = e^{2x} -9$

We have to find the inverse of the equation.

Adding 9 to both sides of above equation, we have

$y+9 = e^{2x} +9-9$

$y+9 = e^{2x}$

Taking logarithms to both sides of above equation, we have

$ln(y+9) = ln(e^{2x})$

$ln(y+9) = 2x$

Dividing by 2 to both sides of above equation, we have

$\frac{1}{2}ln(y+9) = x$

Putting x  = f⁻¹(y) in above equation ,we have

$\frac{1}{2} ln(y+9) = f^{-1}(y)$

Replacing y by x , we have

$\frac{1}{2}ln(x+9) = f^{-1}(x)$

$\frac{1}{2}ln(x+9) = y$ which is the answer.

Answer 2

Answer:

$y=\frac{1}{2} \ln(x+9)$

Step-by-step explanation:

The given function is

$y=e^{2x}-9$

Interchange x and y.

$x=e^{2y}-9$

Solve for y.

$x+9=e^{2y}$

Take logarithm of both sides to base e.

$\ln(x+9)=2y$

Divide both sides by 2.

$y=\frac{1}{2} \ln(x+9)$