The range of the function $f(x)=\frac{sin(\pi[x^{2}+1])}{x^{4}+1}$ where [.] is greatest integer function, is:

Q: The range of the function $f(x)=\frac{sin(\pi[x^{2}+1])}{x^{4}+1}$ where [.] is greatest integer function, is:

To determine the range of the function $ f(x) = \frac{\sin(\pi[x^2 + 1])}{x^4 + 1} $, where $[.]$ denotes the greatest integer function, we need to analyze the behavior of both the numerator and the denominator. 1. **Numerator Analysis:** - The expression inside the sine function is $\pi[x^2 + 1]$. - The greatest integer function $[x^2 + 1]$ takes integer values. Specifically, for any real number $x$, $x^2$ is non-negative, so $x^2 + 1 \geq 1$. - Therefore, $[x^2 + 1]$ can take values like 1, 2, 3, etc., depending on $x$. - The sine function, $\sin(\pi n)$, where $n$ is an integer, is always 0 because $\sin(\pi n) = 0$ for any integer $n$. 2. **Denominator Analysis:** - The denominator $x^4 + 1$ is always positive for all real $x$ because $x^4 \geq 0$, making $x^4 + 1 \geq 1$. 3. **Function Evaluation:** - Since the numerator $\sin(\pi[x^2 + 1])$ is always 0, the entire function $f(x) = \frac{0}{x^4 + 1}$ simplifies to 0 for all real $x$. 4. **Range Determination:** - The function $f(x)$ is constantly 0 for all $x$, meaning the range of the function is $\{0\}$. **Explanation of Incorrect Options:** - **Option 1: [0, 1]** - This option suggests that the function can take any value between 0 and 1, inclusive. However, as shown, the function is constantly 0, and no other values are possible. - **Option 3: [-1, 1]** - This option implies that the function can take any value between -1 and 1. Again, since the function is constantly 0, it does not take any negative values or any positive values other than 0. - **Option 4: {0, 1}** - This option suggests that the function can take the values 0 and 1. However, the function never reaches 1, as the sine term is always 0, making the entire function 0. Thus, the only correct range for the function is $\{0\}$, which is why Option 2 is the correct answer.

Question

Accepted Answer

To determine the range of the function $ f(x) = \frac{\sin(\pi[x^2 + 1])}{x^4 + 1} $, where $[.]$ denotes the greatest integer function, we need to analyze the behavior of both the numerator and the denominator.

1. **Numerator Analysis:**
   - The expression inside the sine function is $\pi[x^2 + 1]$.
   - The greatest integer function $[x^2 + 1]$ takes integer values. Specifically, for any real number $x$, $x^2$ is non-negative, so $x^2 + 1 \geq 1$.
   - Therefore, $[x^2 + 1]$ can take values like 1, 2, 3, etc., depending on $x$.
   - The sine function, $\sin(\pi n)$, where $n$ is an integer, is always 0 because $\sin(\pi n) = 0$ for any integer $n$.

2. **Denominator Analysis:**
   - The denominator $x^4 + 1$ is always positive for all real $x$ because $x^4 \geq 0$, making $x^4 + 1 \geq 1$.

3. **Function Evaluation:**
   - Since the numerator $\sin(\pi[x^2 + 1])$ is always 0, the entire function $f(x) = \frac{0}{x^4 + 1}$ simplifies to 0 for all real $x$.

4. **Range Determination:**
   - The function $f(x)$ is constantly 0 for all $x$, meaning the range of the function is $\{0\}$.

**Explanation of Incorrect Options:**

- **Option 1: [0, 1]**
  - This option suggests that the function can take any value between 0 and 1, inclusive. However, as shown, the function is constantly 0, and no other values are possible.

- **Option 3: [-1, 1]**
  - This option implies that the function can take any value between -1 and 1. Again, since the function is constantly 0, it does not take any negative values or any positive values other than 0.

- **Option 4: {0, 1}**
  - This option suggests that the function can take the values 0 and 1. However, the function never reaches 1, as the sine term is always 0, making the entire function 0.

Thus, the only correct range for the function is $\{0\}$, which is why Option 2 is the correct answer.