Power Functions

Power function helps calculate the space and time required by the manufacturing companies to produce a certain amount of goods designed for a specific purpose. The power functions consider the relationship between variables raised to some power, i.e., an actual number, some constants, and coefficients. This representation can be simple or complex depending on the power function used and its application. Hence, they provide a broader view of algebraic equations, geometry, and various other mathematical operators used to solve a particular problem.

Representation of Power Functions

Many functions are similar to power functions, especially logarithmic functions, but they have minute differences. Thorough knowledge of writing a power function is required to evaluate that.

• A power function is represented by a single term only. It doesn’t consist of multiple terms or equations.
• The single term consists of a variable raised to some power, and that power is a real number value.
• The variable has a coefficient that cannot be zero.
• The single term may have some constants. The combination of all these makes a power function used to represent complex formulae.
• The power functions have specific properties depending on the type and the real number representing the power value.
• According to these properties, specific graphs of a power function are generated that provide overall details about the relationship of terms.
• The basic representation of a power function is given by f(k) = ak^b, where k is the variable whose function is represented, b is the power value that should be a real number. A is the coefficient of k that cannot be 0.

Power Functions Examples

The domain of real numbers is very fast. Hence, there are various ways of representing a power function. This increases the domain of power functions and properties that provide a particular value for the object under consideration. The representation of a power function is based on certain rules that need to be considered while writing or verifying a power equation.

The examples of a power function provide clarity for understanding the rules associated with them.

• f(k) = -ak^(1/8)
• f(l) = (1/b)l^(1/3)
• f(k) = -1/(k^(-1/7))
• f(l) = 81^7
• f(k) = -99k^(1/2)

In the above examples, every power function is a single term representation consisting of a variable that is the main function value. The variable is raised to some power of a real number, and there are non-zero coefficients associated with the variable. Using a power function is to simplify the equations and relationships in mensuration and complex quadratic geometry.

Different Forms of Power Functions

The basic idea of power functions is to provide a single term representation of variables, constants, power value, and coefficients to solve simple and complex equations. There are various categories in which the power function is divided based on the real number value of the power associated with the variable. This covers various domains ranging from architecture to astronomy, providing solutions to complex graph equations.

The basic representation of a power function is given by f(k) = ak^b, where b is the real number value of the power function raised to variable k, and a is the coefficient associated with the variable k.

Based on the value of a and b, the specific forms of power functions are

• Constant Function: Here, the value of power, i.e., b = 0. Therefore, the value of f(k) = a
• Linear Function: Here, the value of power, ie., b = 1. If the value of constant is 1, then, f(k) = k else, f(k) = a*k
• Quadratic Function: Here, the value of power, i.e., b = 2. Therefore, the value of f(k) = ak^2 and if a = 1, then f(k) = k^2
• Cubic Function: Here, the value of power, i.e., b = 3. Therefore, the value of f(k) = ak^3 and if a = 1, f(k) = k^3
• Reciprocal Function: In this type of function, the value of power, i.e., b = -1, -2, -3,….. If the value of b = -1 the f(k) = a/k, if b = -2 then f(k) = a/k^2, if b = -3 then f(k) = a/k^3, and so on.
• Root Function: This type contains square root function, cube root function, etc. The basic idea is to include root values in the power value. For square root, b = 1/2 then f(k) = ak^(1/2), for cube root, b = 1/3 then f(k) = ak^(1/3), and so on.

Rules to Design Graph of Power Functions

The most interesting part of a power function is designing its graphs and finding values of complex equations. Several points should be noted to design a graph using a power function.

• The graph’s symmetry is an important point, i.e., the graph should be symmetrical about the reference point or line.
• The endpoints or end behaviour of the power function are important.
• The graph depends on the nature of the power function, i.e., whether the function is even or odd.
• The graph needs timely transformations for precise values.
• Cross-verify the results by using the end behaviours.

Conclusion

The power functions study material briefly introduces the functions used while solving mensuration problems. The representation of a power function is based on several criteria. It defines a graph form that is used to represent a complex geometry that can be parabolic, hyperbolic, etc. In this way, it covers real numbers, complex equations, geometry, and much more. The graphs generated using a power function provide the relationship between the variable and the associated power value. Hence, it is used to predict the domain range and its boundary values.