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Troyanec [42]
4 years ago
7

Find inverse function of: g(x) = 4x/3-x

Mathematics
1 answer:
hjlf4 years ago
8 0

\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}

<em><u>Solution:</u></em>

<em><u>Given that we have to find the inverse function</u></em>

g(x) = \frac{4x}{3-x}

If a function f(x) is mapping x to y, then the inverse function of f(x) maps y back to x

y=\frac{4x}{3-x}

\mathrm{Interchange\:the\:variables}\:x\:\mathrm{and}\:y

x=\frac{4y}{3-y}

<em><u>Now solve the above expression for "y"</u></em>

x=\frac{4y}{3-y}\\\\\mathrm{Multiply\:both\:sides\:by\:}3-y\\\\x\left(3-y\right)=\frac{4y}{3-y}\left(3-y\right)

x(3-y) = 4y

\mathrm{Expand\:}x\left(3-y\right):\quad 3x-xy

3x-xy = 4y

\mathrm{Subtract\:}3x\mathrm{\:from\:both\:sides}\\\\3x-xy-3x=4y-3x\\\\\mathrm{Simplify}\\\\-xy=4y-3x\\\\\mathrm{Subtract\:}4y\mathrm{\:from\:both\:sides}\\\\-xy-4y=4y-3x-4y\\\\\mathrm{Simplify}\\\\-xy-4y=-3x\\\\

\mathrm{Factor\:out\:common\:term\:}y\\\\-y(x+4) = -3x

\mathrm{Divide\:both\:sides\:by\:}-\left(x+4\right)\\\\\frac{-y\left(x+4\right)}{-\left(x+4\right)}=\frac{-3x}{-\left(x+4\right)}\\\\\mathrm{Simplify}\\\\y = \frac{-3x}{-(x+4)}\\\\\text{Cancel the negative sign in numerator and denominator }\\\\y = \frac{3x}{(x+4)}

\text{ Replace y with } g^{-1}(x)

g^{-1}(x) = \frac{3x}{x+4}

Thus we have got,

\mathrm{Inverse\:of}\:\frac{4x}{3-x} \text{ is } \frac{3x}{x+4}

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