What is a Power Function? 

A power function is a type of single-term function. Its definition states that it is a variable containing a base value raised to a constant value acting as an exponent. This variable may also have a coefficient. For instance, the area of a circle can be given as: 

$A\left(r\right)=\pi r^2$

The above is an example of a power function. The coefficient here is π, which precedes the variable r that is raised to an exponent value of 2.  

Below are two more power functions: 

$$f\left(x\right)=k{x}^a$$$$y\left(x\right)=k{x}^a$$

Here k≠0, which means that it represents a non-zero coefficient. Also, the power 'a' belongs to the real number set (a € R), which means that 'a' represents a real number.

Bird Population: A Real-World Example 

Suppose a species of birds are growing on a particular island. Consider the polynomial given below:  

$P\left(t\right)=-0.5t^3+96t+810$

If the calculations are made starting from 2009,'t' is the input value for the polynomial such that it represents the upcoming years after the starting year. P(t) is a polynomial function of degree 3, used to calculate the maximum number of birds and the year this peak will occur. This model can be used to dictate when the population will vanish from the island as well.  

Other Similar Types of Functions 

There are so many parent functions similar to power functions. For example constant, exponential, linear, quadratic, cubic, reciprocal, square root function, and many more. Some parent functions with their function form have been specified below. 

• Constant function:z=a. 
• Exponential function: y=b^x 
• Linear function: z=y. 
• Quadratic function: x=y² 
• Cubic function: x=y³ 
• Reciprocal function z=2y, x=3z. 
• Square root function: y=z. 

It is important to note that an exponential function differs from a power function in that an exponential function has a variable in its power or exponent, whereas a power function has a variable in its base. Therefore, an exponential function can take on a range of exponents, even a fractional power. For example, z=3^x is an exponential function while z=x³ is a power function. 

Types of Power Functions 

The two types of power functions are even power functions and odd power functions.  

Even Functions: A positive coefficient has been caused by the even power functions. For example, x=2y² 

Odd Functions: A negative coefficient has been caused by the even power functions. For example, x=-2y². 

Process of Finding Power Function 

For finding out the power functions, remember some points as specified below: 

• Substitution of given points into the general form defined by power functions. 

y=kx^a, is the general form. 

• Retaining has been done of either k and a. 
• After the determination of values put back into general form. 

Let us understand this concept with few examples given below. 

Substitution of given points into the general form defined by power functions: 

$\left(2,16\right)=\left(x,y\right)$

Substitute in above mentioned general form:

$16=k{(2)}^a$----- A

$16\times 2^{-a}=k$

$\left(3,54\right)=\left(x,y\right)$

$54=k{(3)}^a$------ B

$54\times 3^{-a}=k$

Equating both sides of expression (right hand side) 

$16\times 2^{-a}=54\times 3^{-a}$

$8\times 2^{-a}=27\times 3^{-a}$

$2^3\times 2^{-a}=3^3\times 3^{-a}$

$2^{3-a}=3^{3-a}$

This will only be true when both sides give the result as one. Clearly explaining that result in the form of 0 has been given by 3-a. therefore, a=3. 

Retaining has been done of either k and a. 

Now that we know the value of a, substituting it in equations A and B can result in deriving the value of k as ‘2’.  

After the determination of the values put it back into the general form, 

Equation:  Y=2x³ 

Properties of Power Function 

• Identification of power function has been done by recognizing a single form. 
• Always apply oddly and even functions properties as per requirement. 
• Utilization of general form has been done by the user for getting output. 

For Even Power Functions 

For Odd Power Functions

Let us take conceptual example for understanding the property of power function. 

Problem

Whenever we used 2 power functions, it will return power function back or not. If yes, explain it. 

Solution: 

Two power functions have been considered here as:f(x)=mx^p, g(x)=nx^q. 

Real number coefficient = m, n 

Exponents = p, q 

After the multiplication of these two functions, we get an output as: 

$f\left(x\right)\cdot g\left(x\right)=m{x}^p\left(n{x}^q\right)$

$mn{x}^{p+q}$

Let us see mn=k and p+q=a  

Therefore we get as in result:

$f\left(x\right)\cdot g\left(x\right)=k{x}^a$

Because of a real number coefficient = m, n; mn and p+q also become real numbers. We can see that in the result, a power function is derived in return. 

So the property proves that whenever 2 power functions are within a product, will return a power function back or not has been satisfied.  

Power Function in C  

Mainly in C language, exponentiation value has been calculated by using pow( ) function. 

Power function in c language has been declared as: 

double pow (double x, double y); 

{ 

   If(y == 0) 

        return 1; 

   return x * pow(x, y – 1); 

} 

Base case of the algorithm: If the exponent is ‘0’ then then result is always 1. 

Otherwise, multiply x, y times and return the result  x^y. 

End Behaviour of Power Functions 

The end behavior of a power function revolves around the idea of infinity. To understand the concept, a graph is drawn for the following power functions. 

$f\left(x\right)=x^2,g\left(x\right)=x^4,h\left(x\right)=x^6, k\left(x\right)=x^8\text{, and} p\left(x\right)=x^{10}$

These power functions are raised to an even number. The five power function graphs all follow a similar curve with a few exceptions. With the increase in power, the graphs flatten close to the origin and take a sharp turn farther out from the origin.  

The x can approach positive infinity and negative infinity for these functions. Therefore, the symbol x→∞ represents the behavior of variable x such that it increases without limits.  

Context and Applications 

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for 

• Bachelor in Mathematics 
• Master’s in Mathematics