Let `f : [-(pi)/(2), (pi)/(2)] rarr [3, 11]` defined as `f(x) = sin^(2)x + 4 sin x + 6`. Show that f is bijective function.

Explanation

To show that the function \( f: [-\frac{\pi}{2}, \frac{\pi}{2}] \to [3, 11] \) defined by \( f(x) = \sin^2 x + 4 \sin x + 6 \) is bijective, we need to demonstrate that it is both one-to-one (injective) and onto (surjective).

### Step 1: Show that \( f \) is one-to-one (injective)

To prove that \( f \) is one-to-one, we need to show that if \( f(a) = f(b) \), then \( a = b \) for any \( a, b \in [-\frac{\pi}{2}, \frac{\pi}{2}] \).

1. Start by rewriting \( f(x) \):
\[
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