If `f(x)=1/(x^(2)+1)` and `g(x)=sinpix+8{x/2}` where {.} denotes fractional part function then the find range of `f(g(x))`

Explanation

To find the range of the composite function \( f(g(x)) \) where \( f(x) = \frac{1}{x^2 + 1} \) and \( g(x) = \sin(\pi x) + 8\left\{ \frac{x}{2} \right\} \), we will follow these steps:

### Step 1: Determine the range of \( g(x) \)

The function \( g(x) \) is composed of two parts: \( \sin(\pi x) \) and \( 8\left\{ \frac{x}{2} \right\} \).

1. **Range of \( \sin(\pi x) \)**:
- The sine function oscillates between -1 and 1.
- Therefore, \( \sin(\pi x) \) will take values in the interval \([-1, 1]\).

2. **Range of \( 8\left\{ \frac{x}{2} \right\} \)**:
- The fractional part function \( \left\{ \frac{x}{2} \right\} \) takes values in the interval \([0, 1)\).
- Thus, \( 8\left\{ \frac{x}{2} \right\} \) will take values in the interval \([0, 8)\).

Combining these two parts:
- The minimum value of \( g(x) \) occurs when \( \sin(\pi x) = -1 \) and \( 8\left\{ \frac{x}{2} \right\} = 0 \), giving \( g(x) = -1 + 0 = -1 \).
- The maximum value of \( g(x) \) occurs when \( \sin(\pi x) = 1 \) and \( 8\left\{ \frac{x}{2} \right\} \) approaches 8, giving \( g(x) = 1 + 8 = 9 \).

Thus, the range of \( g(x) \) is \([-1, 9)\).

### Step 2: Determine the range of \( f(g(x)) \)

Next, we need to find \( f(g(x)) \):
\[ f(g(x)) = f(y) = \frac{1}{y^2 + 1} \]
where \( y = g(x) \).

1. **Evaluate \( f(y) \)**:
- The function \( f(y) \) is defined for all real numbers \( y \).
- The expression \( y^2 + 1 \) is always greater than or equal to 1 (since \( y^2 \geq 0 \)).
- Therefore, \( f(y) \) will always be positive and will take values in the interval \( (0, 1] \).

2. **Find the minimum and maximum of \( f(y) \)**:
- The minimum value of \( f(y) \) occurs when \( y^2 \) is maximized. The maximum value of \( y \) from the range of \( g(x) \) is just below 9.
- Thus, \( f(9) = \frac{1}{9^2 + 1} = \frac{1}{81 + 1} = \frac{1}{82} \).
- The maximum value of \( f(y) \) occurs when \( y = 0 \), giving \( f(0) = \frac{1}{0^2 + 1} = 1 \).

### Conclusion

Combining these results, the range of \( f(g(x)) \) is:
\[ \left( \frac{1}{82}, 1 \right] \]