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Question
Verify that f and g are inverse functions of each other, where f(x) = `(x + 3)/(x - 2)`, g(x) = `(2x + 3)/(x - 1)`
Solution
f(x) = `(x + 3)/(x - 2)`, g(x) = `(2x + 3)/(x - 1)`
Replacing x by g(x), we get
f[g(x)] = `("g"(x)+3)/("g"(x)-2)`
= `(((2x + 3)/(x - 1)) + 3)/(((2x + 3)/(x - 1)) - 2)`
= `(2x + 3 + 3x -3)/(2x + 3 - 2x +2)`
= `(5x)/5`
= x
g(x) = `(2x+3)/(x-1)`
Replacing x by f(x), we get
g[f(x)] = `("2f"(x) + 3)/("f"(x) - 1)`
= `(2((x + 3)/(x - 2)) + 3)/(((x + 3)/(x - 2)) - 1)`
= `(2x+6+3x-6)/(x+3-x+2)`
= `(5x)/5`
= x
Since, f [g(x)] = x and g[f(x)] = x.
∴ f and g are inverse functions of each other.
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