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Let's start by understanding what a polynomial function is. A polynomial function is an expression made up of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In simpler terms, it's a sum of various terms where each term involves a variable raised to a power. For instance, the function given in the exercise, \[ f(x) = x^3 + x^2 - x - 1 \]is a polynomial function. Here, we have four terms:

• \(x^3\) which is the cubic term
• \(x^2\) which is the quadratic term
• \(-x\) which is the linear term
• \(-1\) which is the constant term

Understanding polynomial functions is crucial because they appear frequently in both theoretical and applied mathematics. They can represent a wide range of phenomena and are the building blocks for higher-math concepts.

Function substitution is a method used to evaluate functions at specific points. It's about replacing the variable in the function with a given value. Let's look at an example from the exercise. We need to find \[ f(1) \]To do this, we substitute 1 for \(x\) in the polynomial function: \[ f(1) = 1^3 + 1^2 - 1 - 1 \]Then, we perform the calculations step by step: \[ f(1) = 1 + 1 - 1 -1 = 0 \]We repeat similar steps for other values like \[ f(-1) \] and \[ f\left(\frac{1}{2}\right) \], substituting -1 and \(\frac{1}{2}\) for \(x\) respectively, and then simplify. This process is fundamental in mathematics as it allows you to evaluate the function at any point to understand the behavior and value of the function there.

Function evaluation is the process of calculating the result of a function for certain inputs. You replace the variable in the function with a specific number or expression. This is what was demonstrated in each step of the exercise:

• For \(f(1)\), we substituted 1 into the function and obtained 0.
• For \(f(-1)\), we substituted -1 into the function and also obtained 0.
• For \(f\left(\frac{1}{2}\right)\), we used \(\frac{1}{2}\) and simplified it step by step to get \(-\frac{9}{8}\).

Function evaluation helps to understand the properties, roots, and overall behavior of the function. It is very practical in solving both pure and applied mathematical problems. Whether you're predicting population growth, calculating compound interest, or analyzing physical phenomena, being adept at function evaluation is invaluable.