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The function \( g(x) = \sqrt{4 - x^2} \) is a great example of a semi-circle function. This function is a specific form of the circle equation \( x^2 + y^2 = r^2 \), where \( r \) represents the radius of the circle. For the function \( g(x) \), we focus on the expression \( \sqrt{r^2 - x^2} \), indicating the upper half of the circle. In this case, \( r = 2 \), which means the function represents the upper semicircle of a circle centered at the origin \( (0,0) \) with a radius of 2. Understanding semi-circle functions helps you visualize geometric scenarios and solve problems involving symmetry and partial circular shapes. Connect the dots: positive square roots mean that we are dealing with values above the x-axis, hence the term 'semi-circle'! Remember, the complete circle would have included \( -\sqrt{4 - x^2} \) too.

The domain and range of the function \( g(x) = \sqrt{4 - x^2} \) provide essential information on where the function exists and what values it can take. First, let's talk domain. The domain of a function is the set of all possible inputs (x-values) that give valid outputs. For \( g(x) \), the inside of the square root, \( 4 - x^2 \), must be non-negative because square roots of negative numbers aren't real. This condition is fulfilled for \( -2 \leq x \leq 2 \).Now, let's focus on the range, the set of possible output values (y-values). The square root function only produces non-negative results, meaning that \( g(x) \) is always \( \geq 0 \). The top of the semicircle gives us the maximum y-value, equating to the radius of 2. Hence, our range is \( 0 \leq y \leq 2 \).These constraints ensure that when graphing, you'll capture the entire semi-circle and not go beyond the bounds established by the domain and range.

When graphing the function \( g(x) = \sqrt{4 - x^2} \), it's crucial to follow the identified domain and range. Start with plotting fundamental points within the interval \( -2 \leq x \leq 2 \). - Begin with the endpoints \((-2, 0)\) and \((2, 0)\), as these define edges of the semicircle on the x-axis.- The point \((0, 2)\) represents the top of the semicircle because at \( x=0 \), \( y=\sqrt{4}=2 \).- For simple additional points, try \( x = -1 \) and \( x = 1 \) to see that \( y = \sqrt{3} \approx 1.73 \).After plotting these points on the coordinate system, connect them with a smooth, curved line that represents the semicircular shape. Making sure your graph starts and ends at the x-axis will visually confirm the semi-circle with radius 2. Graphing functions involves not only plotting key points but also understanding the geometry like how symmetry works. This helps in accurately depicting figures like semi-circles on a graph.