Let `A={x in R|-1lt=xlt=1}` and let `f: A->A ,\ g: A->A` be two functions defined by `f(x)=x^2` and `g(x)=sin(pix)/2` . Show that `g^(-1)` exists but `f^(-1)` does not exist. Also, find `g^(-1)` .

Explanation

To solve the problem, we need to analyze the functions \( f(x) = x^2 \) and \( g(x) = \frac{\sin(\pi x)}{2} \) defined on the interval \( A = \{ x \in \mathbb{R} \mid -1 < x < 1 \} \).

### Step 1: Analyze the function \( f(x) = x^2 \)

1. **Determine if \( f(x) \) is one-to-one (injective)**:
- A function is one-to-one if it never takes the same value twice.
- For \( f(x) = x^2 \):
- \( f(-1) = 1 \)
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