Question

H(x) = 2^x
Solve the equation h^-1 (x) =0.5​

Answer 1

The answer is

`√(2) \: \: or \: \: 1.414`

Explanation:

`h(x) = {2}^(x)`

To solve the equation we must first find h-¹(x)

To find h-¹(x) equate h(x) to y

That's

`y = {2}^(x)`

Next interchange the terms

x becomes y and y becomes x

That's

`{2}^(y) = x`

Make y the subject

Take logarithm to base 2 to both sides

That's

`log_(2)( {2} )^(y) = log_(2)x \\ y log_(2)2 = log_(2)(x)`

But

`log_(2)(2) = 1`

`y = log_(2)(x)`

So we have

`{h}^( - 1) (x) = log_(2)(x)`

Now we can solve the equation

We have

`log_(2)(x) = 0.5`

Convert the logarithm into exponential form using the fact that

`log_(a)(x) = b \: \: \cong \: \: x = {a}^(b)`

So we have

`x = {2}^(0.5)`

But

`{2}^(0.5) = √(2)`

So we have the final answer as

`√(2) \: \: or \: \: 1.414`

Hope this helps you