91Ó°ÊÓ

When we talk about arithmetic combinations of functions, we often deal with two or more functions that are combined using basic arithmetic operations — addition, subtraction, multiplication, or division. In this exercise, we're focusing on addition.

Here's a step-by-step breakdown of how you combine the functions:

• Start by evaluating each function separately at a given input.
• Then, perform the specified arithmetic operation — in this case, addition.

Specifically, for functions like \(f(x) = x^2 + 1\) and \(g(x) = x - 4\), evaluating at \(x = 1\) helps us find \(f(1)\) and \(g(1)\).

Once we have the results for \(f(1)\) and \(g(1)\), we simply add them together to get \((f + g)(1)\). This approach allows you to find the combined effect of two functions for a specific input, showcasing how they work together.

Substitution is a fundamental concept in mathematics and plays a crucial role when evaluating functions. It's a simple yet powerful method to replace the variable in a function with a specific number to compute its value.

Here's how substitution works in our exercise:

• Identify the function you need to evaluate. For example, look at \(f(x) = x^2 + 1\) or \(g(x) = x - 4\).
• Replace the variable x in the function with the given input value. For instance, with \(x = 1\), you substitute 1 anytime you see x in your function.

This operation allows us to find specific values like \(f(1)\) or \(g(1)\). Understanding substitution helps in dealing with various function problems as it gives you concrete values to work with.

By practicing this technique, you can confidently tackle a variety of function evaluations and see how functions behave with different inputs.

Polynomial functions are a type of mathematical function that involve terms consisting of a variable raised to whole number powers, like x,x^2, x^3, and so on. In our exercise, \(f(x) = x^2 + 1\) is a classic example of a polynomial function.

Some features of polynomial functions include:

• They can have one or more terms. Here, x^2 and 1 are the terms in \(f(x)\).
• The highest power of x is known as the degree of the polynomial. For example, \(f(x)\) has a degree of 2,since the term with the highest power is x^2.

Polynomial functions like \(f(x) = x^2 + 1\) are smooth and continuous, making them easy to graph and understand. They are foundational in algebra and calculus.

Remember, each term in a polynomial can be calculated by substituting the variable with a specific number, which makes evaluation straightforward. These functions are algebraic expressions that help us decode numerous real-world and theoretical problems.