Why the function is X E R?

Question

1) (ii) Find the range of values of k for which the equation e^x
– 5 = k has no solutions. [1] Ans k less than or equal to 5      ( Please explain how to get the answer)

 

2) Function g is such that g(x) = 2 + 4ln x for x > 0, find the domain of inverse of g.

I know range of g(x) = domain of inverse g(x), why is it the answer given is X E R? how do u get that?

Answer 1

1)

 

e^x - 5 = k

 

 

Add 5 to both sides of the equation.

 

e^x = k + 5

 

 

e^x  is always positive.  Set the following inequality.

 

k + 5 < 0

 

 

Solving for k,

 

k < -5

 

This range of k will have no solutions for x.

 

 

 

2)

 

g(x) = 2 + 4lnx

 

 

To find inverse, we switch places with x and y.

 

x = 2 + 4ln(y)

 

 

Solving for y,

 

x - 2 = 4ln(y)

 

x - 2 = ln(y⁴)

 

 

e^{(x - 2)} = y⁴

 

 

e^x / e² = y⁴

 

⁴√(e^x) / ⁴√(e²) = y = g^-1(x)

 

e^x/4 / e^1/2 = g^-1(x)

 

e^x/4 > 0

 

 

Domain is all real numbers.  A negative or positive exponent will make the result greater than zero.