Examples of constant function in the following topics:
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- Functions can either be constant, increasing as x increases, or decreasing as x increases.
- In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.
- A function is a constant function if f(x)=c for all values of x and some constant c.
- Example 1: Identify the intervals where the function is increasing, decreasing, or constant.
- Identify whether a function is increasing, decreasing, constant, or none of these
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- A rational function is one such that f(x)=Q(x)P(x), where Q(x)≠0; the domain of a rational function can be calculated.
- A rational function is any function which can be written as the ratio of two polynomial functions.
- Any function of one variable, x, is called a rational function if, and only if, it can be written in the form:
- Note that every polynomial function is a rational function with Q(x)=1.
- A constant function such as f(x)=π is a rational function since constants are polynomials.
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- This is accomplished by multiplying either x or y by a constant, respectively.
- Multiplying the entire function f(x) by a constant greater than one causes all the y values of an equation to increase.
- where f(x) is some function and b is an
arbitrary constant.
- Multiplying the independent variable x by a constant greater than one causes all the x values of an equation to increase.
- where f(x) is some function and c is an arbitrary constant.
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- A translation moves every point in a function a constant distance in a specified direction.
- To translate a function vertically is to shift the function up or down.
- where f(x) is some given function and b is the constant that we are adding to cause a translation.
- To translate a function horizontally is the shift the function left or right.
- Where f(x) would be the original function, and a is the constant being added or subtracted to cause a horizontal shift.
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- The function f(x)=ex is a basic exponential function with some very interesting properties.
- The basic exponential function, sometimes referred to as the exponential function, is f(x)=ex, where e is the number (approximately 2.718281828) described previously.
- y=ex is the only function with this property.
- The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
- Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as y=Aekt where A is the number of bacteria present initially (at time t=0) and k is a constant called the growth constant.
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- where a, b, and c are constants and x is the independent variable.
- The constants band c can take any finite value, and a can take any finite value other than 0.
- A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
- When all constants are known, a quadratic equation can be solved as to find a solution of x.
- With a linear function, each input has an individual, unique output (assuming the output is not a constant).
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- A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
- For example, a common equation, y=mx+b, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with x and y as variables and m and b as constants.
- In the linear function graphs below, the constant, m, determines the slope or gradient of that line, and the constant term, b, determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
- The blue line, y=21x−3 and the red line, y=−x+5 are both linear functions.
- Identify what makes a function linear and the characteristics of a linear function
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- The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
- Just as the exponential function with base e arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base e also arises in naturally in many contexts.
- It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions.
- For example, the doubling time for a population which is growing exponentially is usually given as kln2 where k is the growth rate, and the half-life of a radioactive substance is usually given as λln2 where λ is the decay constant.
- The natural logarithm function can be used to solve equations in which the variable is in an exponent.
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- Another easy point to draw is the intersection with the y-axis, as this equals the function value in the point zero, which equals the constant term of the polynomial.
- Conversely, we can easily read the constant term of the polynomial by looking at its intersection with the y-axis if its graph is given (and indeed, we can readily read any function value if the graph is given).
- The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).
- Its constant term is between -1 and 0.
- Its constant term is between 3 and 4.
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- The value of k is meant to adjust the function to compensate for the difference between the expanded form of a(x−h)2 and the general quadratic function ax2+bx+c.
- However, it is possible to write the original quadratic as the sum of this square and a constant:
- Thus, the constant h takes the value −5 and the constant k takes the value −3.
- Solve for the zeros of a quadratic function by completing the square