Question

Let \(f(x)=6 x^3\) and \(g(x)=8 x^3\). Find \((f+g)(x)\). Then evaluate the sum when \(x=2\).

Short answer

\((f+g)(x) = 14 x^3\). The resulting function evaluated at \(x=2\) is 112.

Step-by-step solution

Find the Sum of the Functions

Add the functions \(f(x)\) and \(g(x)\) together. You do this simply by adding the like terms of each function. In this case, the like terms are \(6 x^3\) from \(f(x)\) and \(8 x^3\) from \(g(x)\). So, \((f+g)(x) = 6 x^3 + 8 x^3 = 14 x^3\).

Evaluate the Sum for \(x=2\)

Substitute the value \(x=2\) into the function \((f+g)(x)\). This means replace each \(x\) in the function with the number \(2\). So, \((f+g)(2) = 14*2^3 = 14*8 = 112\).

Key concepts

Algebraic Functions

Algebraic functions are the backbone of algebra and form the foundation for understanding more complex mathematical concepts. They represent a relationship between two variables, typically 'x' and 'y', with 'x' being the independent variable and 'y', often written as 'f(x)', being the dependent variable. These functions can include various operations, such as addition, subtraction, multiplication, division, powers, and roots. In the given exercise, the functions presented are polynomial functions of 'x', specifically cubic functions, denoted as \(f(x)=6x^3\) and \(g(x)=8x^3\).

Understanding algebraic functions involves recognizing the power of the variable 'x'. In our example, 'x' is raised to the third power, making it a cubic term. These functions can shape parabolic curves when graphed. But for the purpose of our exercise, we're interested in combining these functions algebraically to find the sum, a common operation in algebra.

Polynomial Operations

Polynomial operations involve performing arithmetic on polynomials which are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. When combining polynomials, one must understand the concept of 'like terms'. Like terms are terms whose variables (and their exponents) are the same. In polynomial operations, like terms are combined by adding or subtracting their coefficients.

With the given functions in our exercise, \(f(x)=6x^3\) and \(g(x)=8x^3\), we see that both terms are like term since they both have the variable 'x' raised to the power of three. Therefore, to find the sum \((f+g)(x)\), we only need to add the coefficients of these like terms: \(6x^3 + 8x^3 = 14x^3\). This resulting expression is the sum of the two functions, demonstrating the simplicity of polynomial addition.

Function Evaluation

Function evaluation is a procedure where we substitute a numerical value into the place of the variable within a function to find the resultant value of the function. This concept is crucial for understanding how functions behave for specific inputs, and is often used in both simple and complex mathematical problems.

To evaluate the sum function \((f+g)(x)\) when \(x=2\), which we have already established to be \(14x^3\), we replace the 'x' with 2: \((f+g)(2) = 14*2^3\). By calculating the exponent first, \(2^3\), we get 8, followed by multiplying it with 14 to get the final result which is 112. Function evaluation allows us to see the output of the sum function for a given input. This is often a final step in many algebraic problems and forms the bridge between abstract functions and tangible numerical results.