Problem 26 - Algebra A Combined Function

Question

If \(f(x)=\frac{x+8}{2 x-1}\) and \(g(x)=\frac{x-2}{x-5},\) find each function value. $$ f(0) $$

Step-by-Step Solution

Answer

The value of \( f(0) \) is \(-8\).

Step 1: Understand the Function

To find the value of the function \( f(x) \), we need to substitute \( x \) with the given input value. In this problem, we are asked to find \( f(0) \), which means we substitute \( x = 0 \) in the function \( f(x) = \frac{x+8}{2x-1} \).

Step 2: Substitute the Given Value

Substitute \( x = 0 \) into the function. This yields: \( f(0) = \frac{0 + 8}{2(0) - 1} \).

Step 3: Simplify the Expression

Simplify the expression to obtain the value. Calculate the numerator and the denominator separately: the numerator is \( 0 + 8 = 8 \) and the denominator is \( 2(0) - 1 = -1 \). So, \( f(0) = \frac{8}{-1} \).

Step 4: Final Result

Since division of \( 8 \) by \(-1\) results in \(-8\), the value of \( f(0) \) is \(-8\).

Key Concepts

Functions

Functions are a fundamental concept in mathematics that help us represent relationships between variables. Think of a function as a machine where you input a value, and it gives you an output based on a specific rule. Each function is defined by an equation, such as \( f(x) = \frac{x+8}{2x-1} \) in our example.

Here, \( x \) is the input variable — often called the "independent variable" — and \( f(x) \) represents the output or "dependent variable." To evaluate a function, you need to substitute the input value into the function’s equation.

• In our exercise, we are given \( f(x) = \frac{x+8}{2x-1} \) where \( f(0) \) means substituting \( x = 0 \).
• This substitution results in a specific value that illustrates the function's output for that input.

Understanding how to perform this substitution and appreciate the relationship defined by a function is crucial for solving many mathematical problems.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is the foundation for understanding functions and expressions. In algebra, you often perform operations to solve equations or simplify expressions.

In our given problem, the function \( f(x) \) involves an algebraic expression. Here, we utilize basic algebra skills to compute \( f(0) \):

• First, substitute \( x = 0 \) into the function to obtain \( f(0) = \frac{0+8}{2(0)-1} \).
• Next, simplify both the numerator and the denominator using algebraic manipulation, yielding \( f(0) = \frac{8}{-1} \).
• This result illustrates one of algebra's key operations: substitution and simplification.

Mastering these algebraic techniques allows us to solve and simplify functions, a useful skill in mathematics.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. In our exercise, both \( f(x) \) and \( g(x) \) are rational expressions.

A rational expression like \( f(x) = \frac{x+8}{2x-1} \) combines algebraic operations over polynomials, which can be conveniently managed.^

Key points about rational expressions include:

• They can often be simplified by factoring polynomials in the numerator and the denominator.
• Substituting values into rational expressions can yield unique values, as shown by calculating \( f(0) = \frac{8}{-1} = -8 \).
• Rational expressions are particularly sensitive to the values substituted, especially where the denominator might approach zero, which can lead to undefined expressions.

Understanding how to handle rational expressions is vital in algebra, helping in simplifying and evaluating complex equations.