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# How can i find the inverse of this function?

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How can I find the inverse of f^-1?

Guest Oct 9, 2017
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$$f(x)=\sqrt{5-x}+3$$

Changing this to its inverse requires a few steps.

1. Change $$f(x)$$ to $$y$$.

This step is pretty simple. We are switching from function notation to y=-notation. $$f(x)=\sqrt{5-x}+3$$ changes to $$y=\sqrt{5-x}+3$$

2. Interchange x and y

This step is also quite simple; replace all instances of x with y and all instance of y with x.

$$y=\sqrt{5-x}+3$$ changes to $$x=\sqrt{5-y}+3$$

3. Solve for y

This step is the hardest. Transform the equation into the form of y=.

 $$x=\sqrt{5-y}+3$$ Subtract 3 on both sides. $$x-3=\sqrt{5-y}$$ Square both sides to eliminate the square root. $$(x-3)^2=\left(\sqrt{5-y}\right)^2$$ Expand the left hand side knowing that $$(a-b)^2=a^2-2ab+b^2$$. $$x^2-6x+9=5-y$$ Subtract 5 from both sides. $$-y=x^2-6x+4$$ Divide by -1. $$y=-x^2+6x-4$$

4. Consider Whether the Inverse is actually a Function

In this case, it is a function, so we are OK.

TheXSquaredFactor  Oct 9, 2017

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