Question

If the functions are defined as f(x) = √x and g(x) = √(1-x), then what is the common domain of the following functions : f+g, f-g, f/g, g/f, g-f where (f ± g)(x) = f(x) ± g(x), (f/g)(x) = f(x)/g(x)

• A. 0 ≤ x<1
• B. 0<x<1
• C. 0 ≤ x ≤ 1
• D. 0<x ≤ 1

Hint:

When finding the domain of a quotient $f/g$, the domain is $D_f \cap D_g$ excluding points where $g(x) = 0$.

Answer 1

The Correct Option is B

Step 1: Find the domain of $f(x) = \sqrt{x}$. For $\sqrt{x}$ to be real, $x \ge 0$.
Step 2: Find the domain of $g(x) = \sqrt{1-x}$. For $\sqrt{1-x}$ to be real, $1-x \ge 0 \implies x \le 1$.
Step 3: The intersection for $f \pm g$ is $0 \le x \le 1$.
Step 4: For $f/g$, we must also ensure $g(x) \neq 0$. $\sqrt{1-x} \neq 0 \implies x \neq 1$.
Step 5: For $g/f$, we must also ensure $f(x) \neq 0$. $\sqrt{x} \neq 0 \implies x \neq 0$.
Step 6: The common intersection of all conditions is $0<x<1$.