Question

How many \(x\) -intercepts can a function defined on an interval have if it is increasing on that interval? Explain.

Short answer

An increasing function can have at most one x-intercept on an interval.

Step-by-step solution

Understanding the Definition

An increasing function is one where if \(x_1 < x_2\) implies that \(f(x_1) < f(x_2)\) for any two points in the interval.

Define an x-intercept

An \(x\)-intercept is a point where the function value is zero, i.e., \(f(x) = 0\).

Consider the Nature of Increasing Functions

If a function is increasing on an interval, it means it continuously rises. Once it hits zero (if it does), it will not return to zero as it continues to increase. Therefore, it can cross the x-axis at most once.

Conclusion

Thus, an increasing function can have at most one \(x\)-intercept on any given interval.

Key concepts

Increasing Functions

An increasing function is one in which the value of the function rises as the input increases. This means that for any two points on the interval, if one point is to the left of the other (meaning it has a smaller x-coordinate), then the function value at this point is less than the function value at the point to the right.

Mathematically, a function \( f \) is increasing on an interval \( I \) if, for every \( x_1 \) and \( x_2 \) in \( I \), whenever \( x_1 < x_2 \), then \( f(x_1) < f(x_2) \).

This characteristic means that graphically, the line or curve representing the function will always slope upwards as you move from left to right.

Intervals in Functions

An interval in mathematics is a continuous range of values. For example, an interval may be from \( a \) to \( b \), where every number between \( a \) and \( b \) is included.

When we talk about a function being increasing on an interval, we are saying that within this range of values, the function consistently rises and does not fall.

It is important to distinguish between different types of intervals:

• **Closed Interval [a, b]:** Includes both endpoints \(a\) and \(b\).
• **Open Interval (a, b):** Excludes both endpoints \(a\) and \(b\).
• **Half-Open (or Half-Closed) Intervals:** Includes one endpoint and excludes the other, such as \([a, b)\) or \((a, b]\).

Understanding intervals is crucial when determining the behavior of functions over specific ranges.

Function Behavior

The behavior of a function on an interval can significantly affect its characteristics, such as how many x-intercepts it can have.

An x-intercept is where the graph of the function crosses the x-axis. This means that the function value at this point is zero \( (f(x) = 0) \).

For increasing functions, the nature of their behavior ensures that if the function crosses the x-axis (achieves an x-intercept), it will only do so once. Here's why:

• Since the function is always rising, once it crosses from negative to positive values (assuming it does), it won't return to zero.
• This ensures that there's a single point where \( f(x) = 0 \), resulting in only one x-intercept.

This behavior is quite different from other types of functions, such as decreasing functions or quadratic functions, which might have multiple x-intercepts.