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Unformatted text preview: n of G is (−∞, 1). To determine the range of
G, we project the curve to the y -axis as follows:
3 3 2 project right 4 2 1 −1 project left
x 1 1 −1 1 x −1 −1 The graph of G The graph of G Note that even though there is an open circle at (1, 3), we still include the y value of 3 in our range,
since the point (−1, 3) is on the graph of G. We see that the range of G is all real numbers less
than or equal to 4, or, in interval notation: (−∞, 4].
All functions are relations, but not all relations are functions. Thus the equations which described
the relations in Section1.2 may or may not describe y as a function of x. The algebraic representation
of functions is possibly the most important way to view them so we need a process for determining
whether or not an equation of a relation represents a function. (We delay the discussion of ﬁnding
the domain of a function given algebraically until Section 1.5.)
Example 1.4.5. Determine which equations represent y as a function of x:
1. x3 + y 2 = 1
2. x2 + y 3 = 1
3. x2 y = 1 − 3y
Solution. For each of these equations, we solve for y and determine whether each choice of x will
determine only one corresponding value of y .
x3 + y 2
1 − x3
1 − x3
extract square roots
± 1 − x3 1.4 Introduction to Functions 39 √
If we substitute x = 0 into our equation for y , we get: y = ± 1 − 03 = ±1, so that (0, 1)
and (0, −1) are on the graph of...
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