For Which Intervals Is The Function Positive

Of course. Here is a complete, in-depth article on determining the intervals where a function is positive.

For Which Intervals is the Function Positive? A Step-by-Step Guide

Determining the intervals where a function is positive—that is, where its graph lies above the x-axis—is a fundamental skill in algebra and calculus. On top of that, it allows us to solve inequalities, understand the behavior of models in physics and economics, and lay the groundwork for more advanced topics like integration. This guide will walk you through a clear, systematic method for finding these intervals, using detailed examples to illustrate each step.

The Core Concept: What Does "Positive" Mean?

Before diving into the steps, let's clarify the goal. In graphical terms, this means the curve of the function is above the x-axis. A function, f(x), is positive on an interval if, for every x-value within that interval, the corresponding y-value (f(x)) is greater than zero. It's crucial to distinguish this from "increasing," which describes the function's slope, not its position relative to the axis.

The key to solving this problem lies in identifying the points where the function crosses the x-axis. At these points, the function's value is zero. Consider this: these points are called zeros or roots of the function. The zeros divide the x-axis into distinct intervals. Within each of these intervals, the function's sign (positive or negative) will remain constant. This is because a continuous function cannot jump from positive to negative without first passing through zero.

Easier said than done, but still worth knowing.

A Systematic Method for Finding Positive Intervals

Here is a reliable, four-step process you can apply to any continuous function.

Step 1: Find the Zeros of the Function Set the function equal to zero and solve for x: f(x) = 0. These solutions are the critical points that will form the boundaries of your intervals. The methods for finding zeros depend on the type of function:

• Linear: Simple algebra.
• Quadratic: Factoring, completing the square, or the quadratic formula.
• Polynomial of Higher Degree: Factoring by grouping, rational root theorem, etc.
• Rational (a fraction): Set the numerator equal to zero (but remember to exclude values that make the denominator zero, as the function is undefined there).
• Trigonometric/Exponential: Often requires more advanced techniques, but the principle remains the same.

Step 2: Determine the Domain and Identify Vertical Asymptotes For rational functions (fractions), the denominator cannot be zero. Values that make the denominator zero are not part of the function's domain and create vertical asymptotes. These asymptotes, like the zeros, act as boundaries that separate intervals. The function's sign cannot change across an asymptote because the function doesn't exist there; it "jumps" to positive or negative infinity Easy to understand, harder to ignore..

Step 3: Create a Sign Chart This is the most powerful tool for visualizing the problem.

1. Draw a number line.
2. Mark the zeros and any vertical asymptotes (using open circles for asymptotes, as the function is undefined there).
3. These points divide the number line into test intervals.
4. Choose a simple test value from each interval (a number that is not a zero or asymptote) and substitute it into the original function f(x).
5. Determine the sign (positive or negative) of f(x) for that test value. This sign will be the sign for the entire interval.

Step 4: State the Intervals Based on your sign chart, write down the intervals where the function is positive. Use interval notation for clarity. Remember that the endpoints (the zeros) are not included when the question asks for where the function is strictly positive, as the function is zero at those points. On the flip side, if the question allows for non-negative (greater than or equal to zero), you would include them.

Example 1: A Quadratic Function

Let's find the intervals where f(x) = x² - 4 is positive.

Step 1: Find the Zeros. Set f(x) = 0: x² - 4 = 0. Factor: (x - 2)(x + 2) = 0. The zeros are x = 2 and x = -2.

Step 2: Determine the Domain. This is a polynomial, so its domain is all real numbers. There are no vertical asymptotes Not complicated — just consistent. That alone is useful..

Step 3: Create a Sign Chart. The zeros, -2 and 2, divide the number line into three intervals:

1. (-∞, -2)
2. (-2, 2)
3. (2, ∞)

Now, choose a test value from each interval:

• For (-∞, -2), choose x = -3. f(-3) = (-3)² - 4 = 9 - 4 = 5. Positive (+)
• For (-2, 2), choose x = 0. f(0) = (0)² - 4 = -4. That's why Negative (-)
• For (2, ∞), choose x = 3. f(3) = (3)² - 4 = 9 - 4 = 5.

Step 4: State the Intervals. The function f(x) is positive on the intervals (-∞, -2) and (2, ∞) Surprisingly effective..

Visualizing the graph of f(x) = x² - 4, which is a parabola opening upwards, confirms this. It is above the x-axis for x < -2 and x > 2, and below the x-axis between -2 and 2.

Example 2: A Rational Function

Rational functions are more complex due to asymptotes. Let's find where g(x) = (x - 1) / (x + 3) is positive.

Step 1: Find the Zeros. Set the numerator equal to zero: x - 1 = 0 → x = 1. This is the only zero.

Step 2: Determine the Domain and Asymptotes. Set the denominator equal to zero: x + 3 = 0 → x = -3. The function is undefined at x = -3, creating a vertical asymptote. This point must be included on our number line.

Step 3: Create a Sign Chart. The critical points are x = -3 (asymptote) and x = 1 (zero). They divide the number line into three intervals:

1. (-∞, -3)
2. (-3, 1)
3. (1, ∞)

Choose test values:

• For (-∞, -3), choose x = -4. g(-4) = (-4 - 1) / (-4 + 3) = (-5) / (-1) = 5. Positive (+)
• For (-3, 1), choose x = 0. g(0) = (0 - 1) / (0 + 3) = -1 / 3.