In mathematics, a function is a relation that assigns to each element of a set a unique element of another set. A function can be represented graphically as a curve in a coordinate plane, with the independent variable plotted on the x-axis and the dependent variable plotted on the y-axis.
One important property of a function is its sign. A function is said to be positive on an interval if all of its values on that interval are positive. Conversely, a function is said to be negative on an interval if all of its values on that interval are negative.
In this article, we will discuss which functions are positive for the entire interval (3, 2).
Linear Functions
A linear function is a function that can be represented by the equation y = mx + b, where m and b are constants. The graph of a linear function is a straight line.
The sign of a linear function on an interval is determined by the slope of the line. If the slope is positive, then the function is positive on the interval. If the slope is negative, then the function is negative on the interval.
If the slope of a linear function is zero, then the function is constant on the interval. In this case, the function is positive on the interval if and only if the constant is positive.
Quadratic Functions
A quadratic function is a function that can be represented by the equation y = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola.
The sign of a quadratic function on an interval is determined by the sign of the leading coefficient, a. If the leading coefficient is positive, then the function is positive on the interval if the interval is entirely to the right of the vertex of the parabola. If the leading coefficient is negative, then the function is positive on the interval if the interval is entirely to the left of the vertex of the parabola.
If the leading coefficient of a quadratic function is zero, then the function is a linear function.
Rational Functions
A rational function is a function that can be represented by the equation y = P(x)/Q(x), where P(x) and Q(x) are polynomials. The graph of a rational function is a curve that can have both positive and negative values.
The sign of a rational function on an interval is determined by the signs of the numerator and denominator of the function. If the numerator and denominator have the same sign, then the function is positive on the interval. If the numerator and denominator have opposite signs, then the function is negative on the interval.
If the numerator or denominator of a rational function is zero on the interval, then the function is undefined on that interval.
Exponential Functions
An exponential function is a function that can be represented by the equation y = a^x, where a is a constant. The graph of an exponential function is a curve that can have both positive and negative values.
The sign of an exponential function on an interval is determined by the sign of the base, a. If the base is positive, then the function is positive on the entire real line. If the base is negative, then the function is positive on the interval (0, ∞) and negative on the interval (-∞, 0).
Logarithmic Functions
A logarithmic function is a function that can be represented by the equation y = log_a(x), where a is a constant. The graph of a logarithmic function is a curve that can have both positive and negative values.
The sign of a logarithmic function on an interval is determined by the sign of the base, a. If the base is positive, then the function is positive on the interval (0, ∞). If the base is negative, then the function is negative on the interval (0, ∞).
Common Functions That Are Positive for the Entire Interval (3, 2)
The following are some common functions that are positive for the entire interval (3, 2):
- y = x
- y = x^2
- y = x^3
- y = e^x
- y = ln(x)
Applications of Functions That Are Positive for the Entire Interval (3, 2)
Functions that are positive for the entire interval (3, 2) have a variety of applications in mathematics, science, and engineering. Some examples include:
- Modeling population growth: The population of a certain species can be modeled by the function y = e^kt, where k is a constant. If the population is initially positive, then the function will be positive for all time.
- Modeling radioactive decay: The amount of radioactive material present at time t can be modeled by the function y = e^-kt, where k is a constant. If the amount of radioactive material is initially positive, then the function will be positive for all time.
- Modeling the growth of a tumor: The size of a tumor can be modeled by the function y = e^kt, where k is a constant. If the tumor is initially positive, then the function will be positive for all time.
Conclusion
In this article, we have discussed which functions are positive for the entire interval (3, 2). We have also provided some examples of common functions that are positive for this interval and discussed some applications of these functions.