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Question
The domain and range of the real function f defined by f(x) = `(4 - x)/(x - 4)` is given by ______.
Options
Domain = R, Range = {–1, 1}
Domain = R – {1}, Range = R
Domain = R – {4}, Range = {– 1}
Domain = R – {– 4}, Range = {–1, 1}
Solution
The domain and range of the real function f defined by f(x) = `(4 - x)/(x - 4)` is given by Domain = R – {4}, Range = {– 1}.
Explanation:
Given that: f(x) = `(4 - x)/(x - 4)`
We know that f(x) is defined if x – 4 ≠ 0
⇒ x ≠ 4
So, the domain of f(x) is = R – {4}
Let f(x) = y = `(4 - x)/(x - 4)`
⇒ yx – 4y = 4 – x
⇒ yx + x = 4y + 4
⇒ x(y + 1) = 4y + 4
⇒ x = `(4(1 + y))/(1 + y)`
If x is real number, then 1 + y ≠ 0
⇒ y ≠ – 1
∴ Range of f(x) = R {– 1}
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