91Ó°ÊÓ

Polynomial functions are mathematical expressions involving sums of powers of a variable. In our problem, we are given two polynomial functions: \[ f(x) = x^2 + 1 \] and \[ g(x) = 2x - 3 \]. Each term in a polynomial function consists of a coefficient, a variable, and a non-negative integer exponent. For example, in the polynomial \(f(x)\), \(x^2\) is a term where \(x\) is the variable and \(2\) is the exponent. The real number \(1\) is a constant term, which is also considered a polynomial (with exponent \(0\)). Combining these terms gives us the whole polynomial.
Understanding polynomial functions is crucial because they represent a wide range of real-world situations, from simple linear relationships to complex curves.

Combining like terms involves adding or subtracting terms that have the same variables raised to the same powers. This process simplifies polynomial expressions and makes them easier to work with. In the given exercise, after writing down the functions \(f(x) = x^2 + 1\) and \(g(x) = 2x - 3\), we combined them to get: \[ (f + g)(x) = (x^2 + 1) + (2x - 3) \].

To simplify this expression, look for and combine like terms:

• The term \(x^2\) does not have any like terms, so it remains unchanged.
• The linear term \(2x\) is combined with any other like linear terms, but in this case, there are none.
• The constants \(1\) and \(-3\) are combined to give \(-2\).

So, you get: \[ (f + g)(x) = x^2 + 2x + 1 - 3 \] \[ (f + g)(x) = x^2 + 2x - 2 \]. Simplifying by combining like terms helps in reducing the complexity of the polynomial.

Function notation serves as a convenient way to express the actions performed on a variable. A function, denoted as \(f(x)\), implies a specific relationship where each input (\(x\)) is related to an output. In our exercise, we have two functions: \(f(x) = x^2 + 1\) and \(g(x) = 2x - 3\).

The notation \((f + g)(x)\) represents the sum of the outputs of functions \(f\) and \(g\) for a given input \(x\). To find \((f + g)(x)\):

• First, substitute \(x\) into both functions \(f\) and \(g\).
• Then, add the resulting expressions together.

This results in: \[ (f + g)(x) = f(x) + g(x) \]. By understanding function notation, you can easily manipulate and combine multiple functions to explore various mathematical relationships.