Expressions are said to be equivalent if identical inputs al

Expressions are said to be equivalent if identical inputs always produce the same outputs. In term of the chart in problem 5, it appears that the two expressions are equivalent. Explain why a chart cannot be used to prove that two expressions are equivalent. If you were to graph both rules, how would you expect the graphs to compare? Explain why a graph cannot be used to prove that two expressions are equivalent. Symbolically show that the two expressions arc equivalent.

Solution

6. We do not have problem No. 5 , so the question will be answered in a general way.

(a). A chart can contain only a finite number of values in a certain interval. The two expressions may produce identical results for identical inputs in this interval, but the expressions may not produce the same output outside this interval. For example, f(x) = (x² -1)(x-1) and g(x) = (x+1)will produce identical outputs if a chart is prepared for the values of x in the interval (- , 1) or (1,). However , if the interval for x values contains the number 1, then f(x) is not defined at x = 1( as division by 0 is not defined), while g(x) = 2 when x = 1. Thus a chart cannot be used to prove that 2 expressions are equivalent.

( b). If we graph two expressions in a certain domain where the 2 expressions are equivalent, we expect the 2 graphs to be identical/same.

(c ). If we use the example in the para (a)above, then in any domain, excluding x = 1, the graphs of the 2 expressions will be identical. However, the graph of the expression f(x) is discontinuous at x = 1, while the graph of g(x) is a continuous line. Thus a graph cannot be used to prove that 2 expressions are equivalen.

(d). Suppose x(x+2) +3x +2 and x² +5x+2 are 2 different expressions. Since x(x+2) +3x +2 = x²+2x +3x+2 = x² +(2x+3x)+2 = x²+5x +2 , hence we say that the 2 expressions are equivalent and write x(x+2) +3x +2 = x²+5x +2.