Question

If \( f : \mathbb{R} \to \mathbb{R} \), where \( f(x) = \sin x \) and \( g : \mathbb{R} \to \mathbb{R} \), where \( g(x) = x^2 \), then find the range of \( f(x) \) and \( g(x) \).

Hint:

The range of trigonometric functions like sine is limited to the interval \( [-1, 1] \), and the range of quadratic functions like \( x^2 \) is non-negative, i.e., \( [0, \infty) \).

Answer 1

Step 1: Range of \( f(x) = \sin x \).
The sine function \( \sin x \) oscillates between -1 and 1 for all real values of \( x \). Hence, the range of \( f(x) = \sin x \) is: \[ \text{Range of } f(x) = [-1, 1]. \]
Step 2: Range of \( g(x) = x^2 \).
The function \( g(x) = x^2 \) is a parabola that opens upwards, with a minimum value of 0 when \( x = 0 \). Since \( x^2 \geq 0 \) for all \( x \in \mathbb{R} \), the range of \( g(x) = x^2 \) is: \[ \text{Range of } g(x) = [0, \infty). \]
Step 3: Conclusion.
The range of \( f(x) \) is \( [-1, 1] \) and the range of \( g(x) \) is \( [0, \infty) \).