Question

Which among the following options is/are function(s)?

• A. $f:N\to R$ defined by $f\left(x\right)=x^2$
• B. $f:R\to N$ defined by $f\left(x\right)=x^2$
• C. $f:R\to R^{+}$ defined by $f\left(x\right)=x^3$
• D. $f:R^{+}\to R$ defined by $f\left(x\right)=\sqrt{x}$

Answer 1

Answer: (A), (D)

$f:R^{+}\to R$ defined by $f\left(x\right)=\sqrt{x}$

$f:A\to B$ is a function if every element in $A$ has a unique image in $B$ under $f$.

Option $1$:
$f\left(x\right)=\sqrt{x}$
For each $x\in R^{+},$ there is a unique image in $R$.
So, $f\left(x\right)=\sqrt{x}$ is a function.

Option $2$:
$f\left(x\right)=x^2$
$f\left(\frac{1}{2}\right)=\frac{1}{4}\notin N$
So, $f\left(x\right)=x^2$ is not a function.

Option $3$:
$f$ is a function as it satisfies the definition.

Option $4$:
$f\left(x\right)=x^3$
$f(-1)=-1\notin R^{+}$
So, $f\left(x\right)=x^3$ is not a function.