1 Example 1 (a) Let f be the rule which assigns each number to its square. Solution The rule f is given by the formula f(x) = x 2 for all numbers x. Hence the graph of f is the entire parabola y = x 2. Since every blue vertical line crosses the graph of f at exactly one point, it follows that f is a function. The domain of f is the projection of the graph of f onto the x-axis, depicted by the red arrows, which is the entire x-axis. Hence the domain of f is the set . The range of f is the projection of the graph of f onto the y-axis, depicted by the green arrows, which is the upper half of the y-axis. Thus the range of f is the interval [0,  ). Construct the graph of each of the following relations. I f the relation is a function, determine the domain and range of the function from its graph. Use the graph to explain why the relation is or is not a function.

2 (b) Let g be the rule which assigns each positive number to its square. Solution The rule g is given by the formula g(x) = x 2 for x>0. Hence the graph of g is the half of the parabola y = x 2 which lies to the right of the y-axis. Since every blue vertical line crosses the graph of g at exactly one point, it follows that g is a function. The domain of g is the projection of the graph of g onto the x-axis, depicted by the red arrows, which is the right half of the x-axis. Hence the domain of g is the interval (0,  ). The range of g is the projection of the graph of g onto the y-axis, depicted by the green arrows, which is the upper half of the y-axis. Thus the range of g also equals (0,  ).

3 (c) Let x be a number such that there is a square S(x) of side x which lies inside a circle of radius 5. Let h be the rule which assigns each of these numbers x to the area of the square S(x). Solution Recall from Example 1.2C (1) that the largest square which fits inside a circle of radius 5 has side of length Hence h(x) = x 2 has graph the portion of the parabola y = x 2 for 0  x  Since every blue vertical line crosses the graph of h at exactly one point, it follows that h is a function. The domain of h is the projection of the graph of h onto the x-axis, depicted by the red arrows, which is the interval The range of h is the projection of the graph of h onto the y-axis, depicted by the green arrows, which is the interval [0,50].