91Ó°ÊÓ

When you think about a semicircle, imagine cutting a full circle in half. What you're left with is a half-circle, or a semicircle. In this specific function, the equation resembles a semicircle because it fits part of the circle's equation, but only above the horizontal axis.

A circle's equation is usually formatted as \( x^2 + y^2 = r^2 \), where \( r \) is the radius. For our semicircle, replace \( x \) with \( h \), and \( y \) with \( v \), giving us \( h^2 + v^2 = 16 \). The full circle would suggest a repetition below the axis, but since our function is \( v = \sqrt{16 - h^2} \), we are just considering the positive values of \( v \), thus forming the upper half: a semicircle.

• Center of the circle: Origin (0,0)
• Radius: 4 (since \( r^2 = 16 \))
• Graph: Only the top half

Determining the domain and range of a function is crucial for understanding where it is defined and what values it can take. For our semicircle, the domain is the set of all possible \( h \) values. Since we are dealing with a square root function, ensure \( 16 - h^2 \geq 0 \). This inequality simplifies to \( -4 \leq h \leq 4 \). Hence, the domain is the interval \([-4, 4]\).

The range specifies the potential \( v \) values of the function. Here, \( v \) ranges from a minimum of 0 to a maximum of 4, because for the smallest and largest values of \( h \), the \( v \) value is 0. So, when \( h = 0 \), \( v \) reaches its peak value of 4. Thus, you can express the range as \([0, 4]\).

• Domain: \([-4, 4]\)
• Range: \([0, 4]\)

At its core, a square root function translates to the idea of taking the square root of a given expression. It's important to grasp how these functions behave, particularly when applied to expressions that involve other variables. In our function, \( v = \sqrt{16 - h^2} \), the expression under the square root sign must be non-negative. That's why we need \( 16 - h^2 \geq 0 \), as taking square roots of negative numbers doesn't give us real results.

This function describes a curved line, smoothly rising and falling, marking the boundary of a semicircle. The square root function is only defined for specific values of \( h \), allowing us to form the semicircle by grappling with the nature of square roots and their limitations.

• Definition: Involves ensuring a non-negative operand
• Shape: Competent at forming curves, like semicircles