Question

Find the domain and range and sketch the graph of the function \(h\left( x \right) = \sqrt {4 - {x^2}} \).

Short answer

The domain and range of the function are \(\left( { - 2,2} \right)\) and \(\left( {0,2} \right)\).

Step-by-step solution

Define the domain and range of the function

The domain of a function is the set of all inputs for which the formula produces a real number output.

The range of the function \(f\) contains all possible values of \(f\) as \(x\) varies throughout the domain.

Determine the domain and range of the function, and sketch the graph

The function \(h\left( x \right) = \sqrt {4 - {x^2}} \) is defined for all real numbers \(x\).

Let \(y = \sqrt {4 - {x^2}} \). Then

\(\begin{aligned}y = \sqrt {4 - {x^2}}\\{y^2} = 4 - {x^2}\\{x^2} + {y^2} = 4.\end{aligned}\)

Therefore, the graph is the upper half of a circle of radius 2, with its center at the origin.

The domain of the function is shown below:

\(\begin{aligned}\left\{ {x|4 - {x^2} \ge 0} \right\} &= \left\{ {x|4 \ge {x^2}} \right\}\\ &= \left\{ {x|2 \ge x} \right\}\end{aligned}\)

The domain of the function is \(\left( { - 2,2} \right)\).

According to the graph, the range of the function is \(0 \le y \le 2\), or \(\left( {0,2} \right)\).

Thus, the domain and range of the function are \(\left( { - 2,2} \right)\) and \(\left( {0,2} \right)\).