Plot the graph for the inverse function for f(x) = (1/2)^x

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

We are asked to plot the graph of the inverse of the function `f(x)=(1/2)^x` :

We can find the inverse: typically you exchange x and y and then solve for y.

`y=(1/2)^x <==> x=(1/2)^y`

Take log base 2 of both sides.

`log_2 x=log_2 (1/2)^y`

`log_2 x=ylog_2 (1/2)`

`log_2x=-y`

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`y=-log_2 x` is the inverse function

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If `y=log_2 x` is the parent function, this function is reflected over the horizontal axis.

(Note that the graph of `f^(-1)(x)=-log_2 x` will be the reflection of the graph of f(x) across the line y=x.)

The graph of the function in black and the inverse in red.

aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on

`f(x)=(1/2)^x`

By def. of inverse function

`f^(-1)(f(x))=x`    (i)

Let  f(x)=y

`y=(1/2)^x`

`ln(y)=xln(1/2)`

`ln(y)=-x ln(2)`

`x=- ln(y)/ln(2)`        (ii)

Thus from (i) and (ii)

`f^(-1)(y)=-ln(y)/ln(2)`

Its graph is

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