Question

The function \(f\ \R\rightarrow\R\) given by f(x)=7-3x is

• A. not one-one
• B. not onto
• C. even
• D. one-one and onto
• E. odd

Answer 1

The Correct Option is D

The given function is \(f(x) = 7 - 3x\).

We need to determine whether the function is one-one (injective), onto (surjective), even, or odd.

1. Checking if the function is one-one (injective): 

A function is one-one if \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\).

Assume \(f(x_1) = f(x_2)\), i.e., \(7 - 3x_1 = 7 - 3x_2\).

Simplifying: \(-3x_1 = -3x_2\) which gives \(x_1 = x_2\).

Since \(x_1 = x_2\), the function is one-one.

2. Checking if the function is onto (surjective):

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

We solve for \(x\): \(y = 7 - 3x\), so \(3x = 7 - y\), and \(x = \frac{7 - y}{3}\).

Since this is valid for any \(y \in \mathbb{R}\), the function is onto.

3. Checking if the function is even:

A function is even if \(f(-x) = f(x)\) for all \(x \in \mathbb{R}\).

We check: \(f(-x) = 7 - 3(-x) = 7 + 3x\), which is not equal to \(f(x) = 7 - 3x\).

Thus, the function is not even.

4. Checking if the function is odd:

A function is odd if \(f(-x) = -f(x)\) for all \(x \in \mathbb{R}\).

We check: \(f(-x) = 7 + 3x\) and \(-f(x) = -(7 - 3x) = -7 + 3x\).

Since \(f(-x) \neq -f(x)\), the function is not odd.

The function is one-one and onto.