Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f, g,\) and \(h\) are first-degree polynomial functions, then the curve given by \(x=f(t), y=g(t),\) and \(z=h(t)\) is a line.

Short answer

The statement is False. Even though \(f, g,\) and \(h\) each forms a line when considered separately, they might not form a straight line when considered together in three-dimensional space.

Step-by-step solution

Understand the Properties of First-Degree Polynomial Functions

A first-degree polynomial function, also known as a linear function, is a function of the form \(f(x) = mx + c\), where \(m\) and \(c\) are constants and \(x\) is a variable. In the real number domain, a first-degree polynomial function represents a straight line when sketched on a two-dimensional Cartesian coordinate plane.

Apply These Properties to Three-Dimensional Space

If \(f, g,\) and \(h\) are first-degree polynomial functions, then each of them represents a straight line in one-dimensional space. So if we write each individual function in terms of \(t\), we have: \(x=f(t) = mt + c\), \(y=g(t) = nt + d\), and \(z=h(t) = pt + e\), where \(m, n, p, c, d, e\) are constants and \(t\) is a parameter. Now, each of these equations defines a linear relationship between \(t\) and \(x\), \(y\), and \(z\) respectively. Thus, for any given value of \(t\), \(x\), \(y\), and \(z\) will have distinct coordinates in the three-dimensional space.

Reach Conclusion

Even though \(x\), \(y\), and \(z\) each forms a line when considered separately (in one-dimensional space), when they are put together (in three-dimensional space), depending on the specific constants \(m, n, p, c, d, e\), they might form a line or a curved space. Hence, it is not always guaranteed that the curve given by \(x=f(t), y=g(t),\( and \(z=h(t)\) is a straight line in the three-dimensional space. This means the conjecture in the initial statement may not always hold true.