Problem 23 - 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. $$ f(2) $$

Step-by-Step Solution

Answer

The function value is \( f(2) = \frac{10}{3} \).

Step 1: Identify the Function

The function we need to evaluate is given as \( f(x) = \frac{x+8}{2x-1} \). We need to find the value when \( x = 2 \).

Step 2: Substitute the Value of x

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

Step 3: Simplify the Numerator

Compute the numerator: \( 2 + 8 = 10 \), so the numerator is 10.

Step 4: Simplify the Denominator

Compute the denominator: \( 2 \times 2 - 1 = 4 - 1 = 3 \), so the denominator is 3.

Step 5: Compute the Function Value

The value of the function is \( f(2) = \frac{10}{3} \).

Key Concepts

Understanding Rational Functions

Rational functions are expressions that are formed by dividing one polynomial by another. In simpler terms, these functions are fractions where both the numerator and the denominator are polynomials. For example, if you have the expression \( f(x) = \frac{x+8}{2x-1} \), the numerator is \(x + 8\) and the denominator is \(2x - 1\). These are both polynomials. What makes rational functions unique is that the denominator cannot be zero, because division by zero is undefined. This characteristic is important, as it determines the domain of the function, which is the set of all possible x-values you can use in the function. Understanding how to handle both the numerator and the denominator in calculations is key to mastering rational functions.

The Role of Function Substitution

Function substitution is a technique used to evaluate functions at a particular value of \( x \). It involves replacing the variable \( x \) in the function with a given number or expression. This process is straightforward but requires attention to detail. For instance, if you have \( f(x) = \frac{x+8}{2x-1} \) and you want to find \( f(2) \), you substitute every \( x \) you see in the function with 2. It looks like this: \( f(2) = \frac{2+8}{2(2)-1} \). During substitution, be meticulous to avoid mistakes like missing a term or altering the sign. By doing this correctly, you transform the function into a simple arithmetic problem.

Mastering Algebraic Manipulation

Algebraic manipulation is the process of simplifying expressions using algebraic rules and operations. When dealing with rational functions, it often involves simplifying the numerator and the denominator separately before performing the division. For instance, once you substitute \( x = 2 \) into \( f(x) = \frac{x+8}{2x-1} \), you get \( \frac{2+8}{2(2)-1} \). First, address the numerator: \( 2 + 8 = 10 \). Then go to the denominator: \( 2 \times 2 - 1 = 4 - 1 = 3 \). This step-by-step approach ensures clarity and avoids errors. Finally, perform the division: \( \frac{10}{3} \). Proper algebraic manipulation simplifies complex expressions and is a fundamental skill in higher mathematics.