Question

Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 10 - x^2$, then:

• A. $f$ is one-one and onto.
• B. $f$ is one-one but not onto.
• C. $f$ is neither one-one nor onto.
• D. $f$ is onto but not one-one.

Answer 1

Answer: (D)

$f$ is onto but not one-one.

Answer 2

The given function is:

$f(x) = 10 - x^2.$

Step 1: Check for one-one.

A function $f$ is one-one if for $f(x_1) = f(x_2)$, we have $x_1 = x_2$. Assume:

$f(x_1) = f(x_2) \implies 10 - x_1^2 = 10 - x_2^2.$

Simplify:

$x_1^2 = x_2^2 \implies x_1 = \pm x_2.$

Since $x_1 \neq x_2$ in general, the function is not one-one.

Step 2: Check for onto.

A function $f$ is onto if for every $y \in \mathbb{R}$, there exists an $x \in \mathbb{R}$ such that $f(x) = y$. Rearrange:

$f(x) = 10 - x^2 \quad \text{to solve for } x:$ $y = 10 - x^2 \implies x^2 = 10 - y.$

For $x^2 \geq 0$, we require $10 - y \geq 0$, or:

$y \leq 10.$

The function $f(x)$ maps $x \in \mathbb{R}$ to $y \in (-\infty, 10]$. Hence, $f$ is onto.

Conclusion: The function $f(x) = 10 - x^2$ is onto but not one-one.