Question

Enter \(y_{1}=\sqrt{x}, y_{2}=\sqrt{1-x}\) and \(y_{3}=y_{1}+y_{2}\) on your grapher. (a) Graph \(y_{3}\) in \([-3,3]\) by \([-1,3]\) (b) Compare the domain of the graph of \(y_{3}\) with the domains of the graphs of \(y_{1}\) and \(y_{2}\) . (c) Replace \(y_{3}\) by \(y_{1}-y_{2}, \quad y_{2}-y_{1}, \quad y_{1} \cdot y_{2}, \quad y_{1} / y_{2}, \quad\) and \(\quad y_{2} / y_{1}\) in turn, and repeat the comparison of part (b). (d) Based on your observations in \((b)\) and \((c),\) what would you conjecture about the domains of sums, differences, products, and quotients of functions?

Short answer

The graph of \(y_3\) lies within the intersection of the domains of \(y_1\) and \(y_2\), which is [0, 1]. After performing the operations, it's noticed that the domain is mostly defined by the intersection of the domains of the two functions, but with divisions, the domain is further restricted due to the condition that the denominator cannot be zero. Therefore, we can conclude that the domain of a sum, difference, or product of two functions is the intersection of their original domains, but the domain of a quotient is also dependent on where the denominator is not zero.

Step-by-step solution

Understanding the functions

Three functions are given: \(y_{1}=\sqrt{x}\), \(y_{2}=\sqrt{1-x}\) and \(y_{3}=y_{1}+y_{2}\). The domain of \(y_1\) is \([0, \infty)\) because \(x\) should be greater than or equal to 0 under the square root, while the domain of \(y_2\) is \([- \infty, 1]\) because \(1-x\) should be greater than or equal to 0.

Graphing and analyzing \(y_{3}\)

Graph the function \(y_{3}=y_{1}+y_{2}\) in the range \([-3,3]\) by \([-1,3]\). Analyze the graph to identify its domain. The domain should be the intersection of the domains of \(y_1\) and \(y_2\), which is [0, 1].

Comparing domains

Compare the domain of \(y_{3}\) with the domains of \(y_{1}\) and \(y_{2}\). Notice that the domain of \(y_3\) is the intersection of the domains of \(y_1\) and \(y_2\).

Performing operations and repeating comparison

Now replace \(y_{3}\) by \(y_{1}-y_{2}, y_{2}-y_{1}, y_{1} \cdot y_{2}, y_{1} / y_{2}\), and \(y_{2} / y_{1}\) in turn. Graph these functions and compare their domains to the original functions \(y_1\) and \(y_2\). Notice that the domains depend on the restrictions brought by each operation.

Conjecture

Based on the observations in steps 3 and 4, conjecture about the domains of sums, differences, products, and quotients of functions. Sum and difference operations result in a domain that is the intersection of the original function domains. Multiplication does not restrict the domain any further, but the division operation restricts the domain by the condition that the denominator cannot be zero.