Question

\(f(x)=x+\frac{1}{x}, \quad a=1\)

Short answer

The value of the function at \( a = 1 \) is 2.

Step-by-step solution

Identify the Function and the Point

The function given is \( f(x) = x + \frac{1}{x} \) and the point at which it needs to be evaluated is \( a = 1 \).

Substitute the Point into the Function

Replace every \( x \) in the function with 1 to get \( f(1) = 1 + \frac{1}{1} \).

Simplify the Result

Simplify the equation from above to get \( f(1) = 1 + 1 = 2 \).

Key concepts

Understanding Algebraic Functions

Algebraic functions, like the one in our exercise, form the building blocks of algebra and are used to express a relationship between two variables. Here, the function provided is an example of a rational function, specifically, \( f(x) = x + \frac{1}{x} \), which represents a relationship where the output, or function value, is determined by both the input value \(x\) and its reciprocal.

When we talk about algebraic functions, we refer to functions that are made up of operations typical in algebra, such as addition, subtraction, multiplication, division, and root extraction. These functions can become as simple or as complex as needed, and their versatility makes them exceedingly useful in various fields of mathematics and applied sciences.

The Art of Function Simplification

Function simplification is a process by which you reduce a function to its most basic form. Simplifying functions can help make them easier to work with and understand. In the context of our exercise where \( f(x) = x + \frac{1}{x} \), simplification isn't necessarily about making the function 'simpler', since it's already quite straightforward.

However, simplification can refer to the act of streamlining the evaluation process. For instance, upon substituting a particular value for \( x \), like \( a = 1 \), the simplification process involves performing algebraic operations to reach the most reduced form of the result. This means you'll combine like terms, reduce fractions, and carry out any necessary arithmetic to make the function's value clear and unequivocal—as illustrated by the step-by-step solution that simplifies to \( f(1) = 2 \).

Mastering the Substitution Method

The substitution method is one of the most fundamental techniques in algebra. It's the process of replacing variables with numbers or other expressions, as we've done in the given problem; substituting \( x \), with \( a \), which is equal to 1.

This approach isn't confined to simple evaluations; it's also widely used to solve systems of equations, to simplify expressions, and in calculus. When substituting, it is crucial to ensure that the replacement is consistent throughout the expression to avoid any errors.

Why Substitution Works

• It transforms abstract problems into digestible numeracy.
• It allows for the evaluation of functions at specific points.
• It provides a means to simplify complex algebraic expressions by reducing them to more manageable numbers.

The power of substitution is that it can make complicated relationships tangible by narrowing them down to particular values, just as we observed when the abstract \( f(x) \) became the concrete \( f(1) = 2 \).