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Question
Check if the following function has an inverse function. If yes, find the inverse function.
f(x) = `sqrt(4x + 5)`
Solution
f(x) = `sqrt(4x + 5), x ≥ (-5)/4`
Let f(x1) = (fx2)
∴ `sqrt(4x + 5) = sqrt(4x_2 + 5)`
∴ x1 = x2
∴ f is a one-one function.
f(x) = `sqrt(4x + 5)` = y, say(y) ≥ 0
Squaring on both sides, we get
y2 = 4x + 5
∴ x = `(y^2 - 5)/4`
∴ For every y we can get x.
∴ f is an onto function.
∴ x = `(y^2 - 5)/4 = f^-1(y)`
Replacing y by x, we get
`f^-1(x) = (x^2 - 5)/4`
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