Question

The domain and range of the real function f defined by f(x) = ${\displaystyle \frac{4-x}{x-4}}$ is given by ______.

पर्याय

• Domain = R, Range = {–1, 1}

• Domain = R – {1}, Range = R

• Domain = R – {4}, Range = {– 1}

• Domain = R – {– 4}, Range = {–1, 1}

Answer 1

The domain and range of the real function f defined by f(x) = ${\displaystyle \frac{4-x}{x-4}}$ is given by Domain = R – {4}, Range = {– 1}.

Explanation:

Given that: f(x) = ${\displaystyle \frac{4-x}{x-4}}$

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 = ${\displaystyle \frac{4-x}{x-4}}$

⇒ yx – 4y = 4 – x

⇒ yx + x = 4y + 4

⇒ x(y + 1) = 4y + 4

⇒ x = ${\displaystyle \frac{4(1+y)}{1+y}}$

If x is real number, then 1 + y ≠ 0

⇒ y ≠ – 1

∴ Range of f(x) = R {– 1}