Question

Find the inverse function of "TexFormula1" title="f(x) = 9 + \sqrt{4x - 4} " alt="f(x) = 9 + \sqrt{4x - 4} " align="absmiddle" class="latex-formula">
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Answer 1

Answer:

$\displaystyle {f}^{ - 1}(x) = \frac{1}{4} (x - 18x + 85)$

Step-by-step explanation:

we would like to find the inverse of the following function:

$\displaystyle f(x) = 9 + \sqrt{4x - 4}$

to do so substitute y for f(x):

$\displaystyle y= 9 + \sqrt{4x - 4}$

interchange variables:

$\displaystyle x= 9 + \sqrt{4y - 4}$

cancel 9 from both sides:

$\displaystyle \sqrt{4y - 4} = x - 9$

square both sides:

$\displaystyle 4y - 4= x - 18x + 81$

add 4 in both sides:

$\displaystyle 4y = x - 18x + 85$

divide both sides by 4:

$\displaystyle y = \frac{1}{4} (x - 18x + 85)$

substitute back:

$\displaystyle {f}^{ - 1}(x) = \frac{1}{4} (x - 18x + 85)$

and we're done!

Answer 2

Answer:

The inverse is 1/4(x-9)^2+1

Step-by-step explanation:

y = 9+ sqrt(4x-4)

Exchange x and y

x =  9+ sqrt(4y-4)

Solve for y

Subtract 9 from each side

x-9 = sqrt(4y-4)

Square each side

(x-9)^2 = 4y-4

Add 4 to each side

(x-9)^2 +4 = 4y

Divide by 4

1/4(x-9)^2+ 4/4 = 4y/4

1/4(x-9)^2+1 = y

The inverse is 1/4(x-9)^2+1