Question

Which of the following statements are correct regarding the function f(x) = $\sqrt{x}$

• A. Range is $R^{+}$ U {0}
• B. Its graph is the mirror image of y = $x^2$ about y = x for x ≥ 0
• C. f(x) ≥$x^2$ in the interval [0, 1]
• D. f(x) ≤$x^2$, x∈ R

Answer 1

Answer: (A), (B), (C)

f(x) ≥$x^2$ in the interval [0, 1]

A. The given function is f(x) = $\sqrt{x}$ . $\sqrt{x}$ is not defined if x < 0. So its domain is the set of all positive real numbers plus zero. This is not same as set of whole numbers.
B. We know $\sqrt{x}\ge$ 0 its range is $R^{+}$ U {O}, same as its domain.
C. Inverse of f(x) = $\sqrt{x}$ is g(x) = $x^2$ , because

f(g(x)) = $(x^2{)}^{\frac{1}{2}}$ = x if x $\ge$ 0 and

g(f(x)) = $(x^{\frac{1}{2}}{)}^2$ = x if x $\ge$ 0

Since they are inverse of each other, their graphs will be mirror image of each other about y = x.
D. f(x) $\ge$ $x^2$ if x $\in$ [0,1] $\sqrt{x}$ is greater than $x^2$ in the interval (0, 1). For example $\sqrt{\frac{1}{4}}$$>$${\left(\frac{1}{4}\right)}^2$ because $\frac{1}{2}$ $>$ $\frac{1}{16}$ .
E. f(x) $\le x^2$ is not correct, because $\sqrt{x}\ge x^2$ in [0, 1]