(a) The inverse of a function \(f\) is given by \(f^{-1}(x)=\frac{5x - 6}{4 - x},x ≠ 4\).Find the:
function, \(f (x)\)
(b) The inverse of a function \(f\) is given by \(f^{-1}(x)=\frac{5x - 6}{4 - x},x ≠ 4\).Find the:
value of x for which \(f (x) = 5\)
(a) \((f^{-1})^{-1}(x)=f(x)\)
\(f^{-1}(x)=\frac{5x - 6}{4 - x},x ≠ 4\)
Let \(y=\frac{5x - 6}{4 - x}\)
\(=y(4-x)=5x-6\)
\(=4y-xy=5x-6\)
\(=-xy-5x=-6-4y\)
\(=x(-y-5)=-6-4y\)
\(=x=\frac{-6 - 4y}{-y - 5}=\frac{-(6 + 4y)}{-(y + 5)}\)
\(=x=\frac{6 + 4y}{y + 5}\)
\(∴f(x)=\frac{6 + 4x}{x + 5},x≠-5\)
(b) \(f(x)=5\)
\(=\frac{6 + 4x}{x + 5}=5\)
\(=6+4x=5(x+5)\)
\(=6+4x=5x+25\)
\(=4x-5x=25-6\)
\(=-x=19\)
\(∴x=-19\)
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