Question

Consider the following statement. Linear functions always have one x-intercept. Give an example if the statement is true or a counterexample if the statement is not true."

Answer 1

A linear function is a function that can be represented by a straight line. Not all linear functions have only one x-intercept, which can be shown with counterexamples. For example, the equation y = 0 represents a horizontal line with infinitely many x-intercepts.

Step-by-step explanation:

A linear function is a function that can be represented by a straight line on a graph. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The x-intercept is the point where the graph of the line crosses the x-axis, which occurs when y = 0.

If the statement that linear functions always have one x-intercept is true, then all linear functions should have exactly one x-intercept. However, this statement is not true. A counterexample to this statement is the equation y = 0, which represents a horizontal line that intersects the x-axis at infinitely many points. So, there are linear functions that can have more than one x-intercept.

For example, let's consider the equation y = 2x - 4. To find the x-intercept, we set y = 0 and solve for x:

1. 0 = 2x - 4
2. 2x = 4
3. x = 2

So, the equation y = 2x - 4 has one x-intercept at (2, 0).