Question

Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Short answer

The curve is a graph of the function, and the domain and range of function are\(\left( { - 2,\;2} \right)\), and\(\left( { - 1,\;2} \right)\), respectively.

Step-by-step solution

Vertical line test

A graph is said to be the graph of a function of x if and only if no vertical line intersects the curve more than once.

 Step 2: Determine whether the curve is the graph of a function of x

Draw a vertical line as follows.

From the graph, it can be seen that if a vertical line is drawn on the curve, then the line will intersect the graph at most one point.

Therefore, the curve is a graph of the function.

Determine the domain and range of the function

From the graph, it can be observed that the function exists for all the values of\( - 2 \le x \le 2\). Therefore, the domain of the function is\(\left( { - 2,\;2} \right)\).

The function takes all values from \( - 1\)to\(2\). Therefore, the range of the function \(f\)is\(\left( { - 1,\;2} \right)\).

Therefore, the domain and range of function are\(\left( { - 2,\;2} \right)\), and\(\left( { - 1,\;2} \right)\),

respectively.