91Ó°ÊÓ

Adding functions simply means combining the outputs of two functions to create a new function. For the given problem, we are given two functions:

• \( f(x) = x^2 - 9 \)
• \( g(x) = 2x \)

To add these functions together, we use the notation \((f + g)(x)\), which means we sum the values of \(f(x)\) and \(g(x)\).

This can be written exactly as: \((f + g)(x) = f(x) + g(x)\).

Next, plug in the given functions:

• \( f(x) = x^2 - 9 \)
• \( g(x) = 2x \)

Combining these, we get: \( (f + g)(x) = (x^2 - 9) + 2x \).

To simplify the function, we should combine like terms. Like terms are terms that have the same variables raised to the same power.

In our expression \( (x^2 - 9) + 2x \), the terms are:

• \( x^2 \) - the quadratic term
• \( 2x \) - the linear term
• \( -9 \) - the constant term

Since these terms do not have the same variable parts, they cannot be combined further. Thus, the expression \( (f + g)(x) \) simplifies to:

\[ (f + g)(x) = x^2 + 2x - 9 \].

Understanding polynomial functions is key to performing function operations. A polynomial function is an expression consisting of variables, coefficients, and non-negative integer exponents of variables.

For example, the function \( f(x) = x^2 - 9 \) is a polynomial because it contains the variable \( x \), a coefficient, and an exponent. The same applies to the function \( g(x) = 2x \).

When we add two polynomial functions, the result is another polynomial function. In our case:

• \( (f + g)(x) = x^2 + 2x - 9 \)

This resultant function is also a polynomial, composed of terms that include a variable raised to different powers (quadratic and linear terms) and a constant. Understanding these basics will make operations with polynomials easier and will provide a strong foundation for more advanced algebra concepts.