Question

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

Answer 1

Answer:

The answer is

$\sqrt{2} \: \: or \: \: 1.414$

Step-by-step 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

${2}^{0.5} = \sqrt{2}$

So we have the final answer as

$\sqrt{2} \: \: or \: \: 1.414$

Hope this helps you