Question

Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \[ f(x)=5^{x^4+2} \] Then:

• A. \(f\) is one-one but not onto
• B. \(f\) is onto but not one-one
• C. \(f\) is both one-one and onto
• D. \(f\) is neither one-one nor onto

Hint:

If a function contains even powers like \(x^2,x^4,x^6\), always test \(f(x)\) and \(f(-x)\) to check whether the function is one-one.

Answer 1

The Correct Option is D

Step 1: Checking whether the function is one-one.
A function is one-one if different inputs always give different outputs.
Given, \[ f(x)=5^{x^4+2} \] Now observe that \[ (-x)^4=x^4 \] Therefore, \[ f(-x)=5^{(-x)^4+2} \] \[ f(-x)=5^{x^4+2} \] \[ f(-x)=f(x) \] For example, \[ f(1)=5^{1+2}=5^3 \] and \[ f(-1)=5^{1+2}=5^3 \] Since \[ f(1)=f(-1) \] but \[ 1\neq -1, \] the function is not one-one.

Step 2: Check whether the function is onto.
For all real values of \(x\), \[ x^4\geq 0 \] Hence, \[ x^4+2\geq 2 \] Therefore, \[ f(x)=5^{x^4+2}\geq 5^2 \] \[ f(x)\geq 25 \] So the range of the function is \[ [25,\infty) \] But the codomain is \[ \mathbb{R} \] Since values less than \(25\) are not obtained, every real number is not an image of some \(x\in\mathbb{R}\).
Hence, the function is not onto.

Step 3: Final conclusion.
The function is neither one-one nor onto.
Therefore, \[ \boxed{\text{\(f\) is neither one-one nor onto}} \]