Need help with second verify

[KarlieWarliee]

I am stuck on how to simplify the second verify of the inverse equation. Its on the right.. problem 2 of number 4

 

[Prove It]

I am stuck on how to simplify the second verify of the inverse equation. Its on the right.. problem 2 of number 4

When you swap x and y, complete the square so that you can solve for y in terms of x.

 

[KarlieWarliee]

When you swap x and y, complete the square so that you can solve for y in terms of x.

Well I found the equation but then when it says to verify It gets really complicated. Is there a simpler inverse equation I could have come up with? (see picture with my work)

 

[stapel]

(see picture with my work)

That's difficult, because your image is small and sideways, and your writing is faint. I think you mean your post to be as follows:

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Let f (x) = x² - 6x + 8. Graph the function and determine a restriction on the domain that will make f a one-to-one function. Then find its inverse function f ^-1 on this domain. Lastly, verify both compositions:

. . . . .\(\displaystyle \left(f\, \circ\, f^{-1}\right)(x)\, =\, x\, \mbox{ and }\, \left(f^{-1}\, \circ\, f\right)(x)\, =\, x\)

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1. graph [not shown here]

2. make one-to-one:

. . .domain: x < 3

. . .range: y > -1

3. inverse:

. . .y = x² - 6x + 8

. . .0 = x² - 6x + (8 - y)

. . .\(\displaystyle x\, =\, \dfrac{-(-6)\, \pm\, \sqrt{\strut (-6)^2\, -\, 4(1)(8\, -\, y)\,}}{2(1)}\)

. . .\(\displaystyle x\, =\, \dfrac{6\, \pm\, \sqrt{\strut 36\, -\, 32\, +\, 4y\,}}{2}\)

. . .\(\displaystyle x\, =\, \dfrac{6\, -\, \sqrt{\strut 4\, +\, 4y\,}}{2}\)

. . .swap:

. . .\(\displaystyle y\, =\, \dfrac{6\, -\, \sqrt{\strut 4\, +\, 4x\,}}{2} \, \longrightarrow\, \boxed{\, f^{-1}(x)\, =\, \dfrac{6\, -\, \sqrt{\strut 4\, +\, 4x\,}}{2}\,}\)

[table in corner not entirely visible]

["verify" runs off page]​

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Assuming I'm correct, then:

...Is there a simpler inverse equation I could have come up with?

Yes: Try factoring inside the square root, and taking out the common (and squared) factor. Then factor the numerator, and cancel. You'll have something with no fraction, which will definitely be simpler!