H in the equation x 3 y 2 6 y is not a function of x

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rithms, powers, and exponential functions. (h) In the equation x = 3( y 2 – 6) , y is not a function of x , since (rearranging) y = ± [( x + 18)/3] 1/2 , so there are two values of y corresponding to each value of x. If we constrain y to be non-negative, however, then over this restricted range y is a M ATH M ODULE S olutions to Exercises 1
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function of x. If we do not want y to be imaginary (the square root of a negaitve number), then we must also restrict the domain of x so that x ≥ –18. (i) Rearranging, we have y = f ( x ) = 3/4 –1/2 x , a monotonically decreasing linear function, with inverse function x = 3/2 – 2 y. (j) With the constant (vertical) equation x = 7, y is not a function of x , because to the only value of x in the domain of the function ( x = 7) correspond an infinite num- ber of values of y. Of course, x is a (constant) function of y, since the function x = g ( y ) = 7 assigns the same value (= 7) to x for every value of y . (k) y = f ( x ) = 14 is a horizontal constant function of x , since the function assigns the same constant value of y (that is, y = 14) for any value of x. For this reason, how- ever, there is no inverse function x = g ( y ). 2. As the Figure below indicates, this was a semi-trick question. While the equations look somewhat different, simple algebraic manipulation puts them all into the same “standard” form, y = 50 – 2 x , with vertical intercept (0, 50), horizontal intercept (25, 0), and slope –2. Such linear equations are examined more closely in Modules 2 and 3, but here you should notice two points in particular. First, Equation (c) is the inverse function of Equation (b). And second, Equation (a) is identical to Equation (d) except that all of the parameter values have been multiplied by the same constant factor, 2. This fact has an important economic interpretation. If these equations represent a budget line, then a proportional increase or decrease in money income and in all goods’ prices has no effect on the position of the budget constraint and hence on feasible con- sumption possibilities.
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