The Sign of a Function: Definition, Analysis, Graph Behavior, and Applications

1. What Is the Sign of a Function?

The sign of a function refers to whether the function’s output is positive, negative, or zero for different values of x. For any function f(x):

• f(x) > 0 → the function is positive
• f(x) < 0 → the function is negative
• f(x) = 0 → the function crosses or touches the x-axis

2. Why the Sign of a Function Matters

Knowing the sign of a function helps us:

• Understand graph behavior
• Solve inequalities such as f(x) > 0 or f(x) < 0
• Identify intervals where the function is above or below the x-axis
• Analyze roots and turning points

3. Zeros of a Function and Sign Changes

The zeros of a function (solutions to f(x) = 0) divide the domain into intervals. The sign of the function is constant within each interval.

Example:

f(x) = x - 3
Zero: x = 3

Thus:

• x < 3 → f(x) is negative
• x > 3 → f(x) is positive

4. Using a Sign Chart

A sign chart helps determine where a function is positive or negative. Steps:

• Find the zeros of the function
• Divide the number line into intervals
• Pick a test point in each interval
• Evaluate the sign of f(x) at each test point

Example:

f(x) = (x - 1)(x + 2)
Zeros: x = 1, x = -2

Intervals:

• x < -2 → both factors negative → f(x) positive
• -2 < x < 1 → one factor negative, one positive → f(x) negative
• x > 1 → both factors positive → f(x) positive

5. Graph Interpretation of Sign

The sign of a function corresponds directly to its position relative to the x-axis:

• Above the x-axis → f(x) > 0
• On the x-axis → f(x) = 0
• Below the x-axis → f(x) < 0

Example:

f(x) = x^2 - 4
Zeros: x = -2, x = 2

The graph is below the x-axis between -2 and 2, and above it outside this interval.

6. Sign of Rational Functions

For rational functions, the sign depends on both numerator and denominator:

f(x) = (x - 1) / (x + 3)

Zeros and undefined points divide the number line:

• Zero at x = 1
• Undefined at x = -3

Analyze signs of numerator and denominator separately in each interval.

7. Applications of Sign Analysis

• Solving inequalities
• Studying increasing/decreasing behavior
• Analyzing polynomial roots
• Understanding rational function behavior
• Graph sketching and calculus problems

8. More Examples

Example 1:

f(x) = -x
Sign: positive for x < 0, negative for x > 0

Example 2:

f(x) = x(x - 5)
Zeros: 0, 5
Sign: negative between 0 and 5, positive outside

9. Conclusion

The sign of a function reveals where the function is positive, negative, or zero. By analyzing zeros, intervals, and the graph, we can understand the behavior of the function and solve inequalities more effectively. Sign analysis is a fundamental tool in algebra, calculus, and graph interpretation.