Problem 127 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Step-by-Step Solution

Answer

Yes, the statement is true. It is indeed possible for a rational function to have no y-intercept. The rational function \( f(x) = \frac{x}{x}\) is an example of such a function, where the function is undefined at \(x = 0\) and hence has no y-intercept.

Step 1: Understanding the concept of y-intercept

The y-intercept is the point where a function or a graph crosses the y-axis. It's the value of the function when \(x = 0\). In order for a rational function to have no y-intercept, the functional value at \(x = 0\) must be undefined.

Step 2: Considering a rational function without y-intercept

Consider the rational function \( f(x) = \frac{x}{x}\). For all real values of \(x\), \( f(x) = 1\). However, at \(x = 0\), the function is undefined because division by zero is undefined in mathematics.

Step 3: Conclusion

This shows it is indeed possible for a rational function to lack a y-intercept. The function \( f(x) = \frac{x}{x}\) is an example of such a function, where the function is undefined at \(x = 0\) and hence has no y-intercept.