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hjlf
1 year ago
11

Question 19 of 25 f(x)=√x-6. Find the inverse of f(x) and its domain.

Mathematics
1 answer:
Gekata [30.6K]1 year ago
5 0

Answer:

f¯¹(x)=x²+12x+36

Step-by-step explanation:

f(x) =  \sqrt{x}  - 6 \\ substitute \: y \: for \: f {}^{ - 1} (x) \\ y =  \sqrt{x}  - 6 \\ interchange \: x \: and \: y \\ x =  \sqrt{y}  - 6 \\ swap \: the \: side \: of \: the \: equation \\  \sqrt{y }  - 6 = x \\ move \: the \: constant \: to \: the \: right \: hand \: side \\   \sqrt{y}  = x + 6 \\ square \: both \: sides \: of \: the \: equation \\ y = x {}^{2}  + 12x + 36 \\ substitute \: f \frac{}{ -1} (x) \: for \: y \\ f {}^{ - 1} (x) = x {}^{2}  + 12x + 36 \\ domain \: x∈R

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\bf \begin{array}{lccclll}
&\stackrel{gallons}{acid}&\stackrel{acid~\%}{quantity}&\stackrel{acid~gallons}{quantity}\\
&------&------&------\\
\textit{pure acid}&x&1.00&x\\
\textit{20\% sol'n}&6&0.20&1.2\\
------&------&------&------\\\
mixture&y&0.90&0.9y
\end{array}
\\\\\\
\begin{cases}
x+6=\boxed{y}\\
x+1.2=0.9y\\
----------\\
x+1.2=0.9\left( \boxed{x+6} \right)
\end{cases}
\\\\\\
x+1.2=0.9x+5.4\implies x-0.9x=5.4-1.2\implies 0.1x=4.2
\\\\\\
x=\cfrac{4.2}{0.1}\implies x=\stackrel{gallons}{42}
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