Find the power function that the graph of f resembles for large values of |x|, given the function f(x) = (x + 6)2 (x – 2)2

Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 7⁵. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 11³, here, 11 is the base value and 3 is the exponent or power of 11. The value of 11³ is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cx^y where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as

p × p × p × p … n times = pⁿ

Functions

A function can be defined as the set of rules relating to a given set of inputs that provide some possible outputs. Only those expressions are denoted as functions in which there is one output for one input. Can there be two inputs for the same output? Yes. However, there cannot be two outputs for a single input.

Functions can be represented as f(x), g(x), h(x), etc. Here, f(x) is the output for a given value of input in the polynomial. For example, the value of f(x) for x = -2 in the function f(x) = 2x + 20 will be 16. It is obtained by placing the value of x in the expression and solving it.

Multiplying and Dividing functions

In order to multiply or divide two functions, the first requirement is to understand that multiplication and division are basic mathematical multiplication and division. Just the way numbers are multiplied or divided, similarly, polynomials are multiplied and divided. They can be represented as f(x).g(x) for multiplication and f(x)/g(x) for division.

Power Function

The product of a real number, a coefficient, and a variable raised to a fixed real number or a natural number is known as a power function. In simple words, a power function can be denoted as a variable raised to a real number. A power function is represented as y = x^R where R is any real number. For instance, y = x² is a power function, y = 1/x is also a power function, and so on. 

Question: Find the power function that the graph of f resembles for large values of |x|, given the function f(x) = (x + 6)² (x – 2)².

Solution:

First expand the expression on the RHS by using the following formula,

• (a + b)² = a² + b² + 2ab
• (a – b)² = a² + b² – 2ab

f(x) = (x² + 36 + 12x)(x² + 4 – 4x)

Now, multiply both terms,

f(x) = (x⁴ + 4x² – 4x³ + 36x² + 144 – 144x + 12x³ + 48x – 48x²)

f(x) = x⁴ + 8x³ – 18x² – 96x + 144)

As it is clear that the degree of the function is 4. Therefore, the power function that the graph of f resembles for large values of |x| is x⁴.

Similar Problems

Question 1: Given the function f(x) = x⁵ + 56x⁴ – 78x + 2. Find the power function that the graph of f resembles.

Solution:

Since the function given in the question is already expanded. Therefore, there is no requirement of expanding the function.

f(x) = x⁵ + 56x⁴ – 78x + 2

As it is clear that the degree of the function is 5. Therefore, the power function that the graph of f resembles is x⁵.

Question 2: Given the function f(x) = (x + 1)² (x – 1)². Find the power function that the graph of f resembles for large values of |x|.

Solution:

First expand the expression on the RHS by using the following formula,

• (a + b)² = a² + b² + 2ab
• (a – b)² = a² + b² – 2ab

f(x) = (x² + 1 + 2x)(x² + 1 – 2x)

Now, multiply both terms,

f(x) = (x⁴ + x² – 2x³ + x² + 1 – 2x + 2x³ + 2x – 4x²)

f(x) = x⁴ – 2x² + 1)

As it is clear that the degree of the function is 4. Therefore, the power function that the graph of f resembles for large values of |x| is x⁴.

Question 3: Given the function f(x) = (x⁵) (x + 3)². Find the power function that the graph of f resembles for large values of |x|.

Solution:

First expand the expression on the RHS by using the following formula,

(a + b)² = a² + b² + 2ab

f(x) = (x² + 9 + 6x)(x⁵)

Now, multiply both terms,

f(x) = (x⁷ + 9x⁵ + 6x⁶)

As it is clear that the degree of the function is 7. Therefore, the power function that the graph of f resembles for large values of |x| is x⁷.