Problem 24 - 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. $$ g(10) $$

Step-by-Step Solution

Answer

The value of \( g(10) \) is \( \frac{8}{5} \).

Step 1: Understand the Function

We are given the function \( g(x) = \frac{x-2}{x-5} \). To find \( g(10) \), we need to substitute \( x = 10 \) into the function.

Step 2: Substitute the Value

Substitute \( x = 10 \) into the function \( g(x) = \frac{x-2}{x-5} \). This gives us \( g(10) = \frac{10-2}{10-5} \).

Step 3: Simplify the Expression

Simplify \( \frac{10-2}{10-5} \). This results in \( \frac{8}{5} \).

Key Concepts

Function Evaluation

Function evaluation is a key concept in algebra, where we determine the value of a function for a specific input. This process transforms abstract functions into concrete numbers, making them easier to analyze and understand. For example, if we have a function defined as \( g(x) = \frac{x-2}{x-5} \), evaluating this function for \( x = 10 \) means finding out what number results when 10 is plugged into the function instead of \( x \).

• This involves identifying the function, which typically appears in a formulaic form.
• Next, substitute the specified value for the variable \( x \).
• Finally, perform any necessary calculations to find the result.

The purpose of function evaluation is to see how inputs affect outputs, and it's an essential step in understanding how functions model real-world situations.

Substitution Method

The substitution method is a straightforward approach used in evaluating functions and solving equations. When we talk about substituting in mathematics, we are replacing a variable with a given number or another expression.To utilize this method, follow these steps:

• Identify the variable you need to substitute. In typical problems, this is given in the form of \( x = a \), where \( a \) is a specific number.


• Replace every instance of the variable in the function with the specified value. This involves performing precise arithmetic operations wherever the variable appears.

For example, in evaluating \( g(10) \) for the function \( g(x) = \frac{x-2}{x-5} \), the substitution involves plugging \( 10 \) in place of \( x \), resulting in \( \frac{10-2}{10-5} \). This technique simplifies problems by turning abstract expressions into ones that are more manageable.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, making it easier to understand and work with. This entails ensuring that the numerator (the top part of the fraction) and the denominator (the bottom part of the fraction) are as small as possible while still maintaining the same value.Here's how to simplify a fraction:

• First, perform the arithmetic operations in the numerator and the denominator, if necessary.


• After calculating, look for numbers that divide both the numerator and the denominator evenly. These are known as the greatest common divisors (GCD).
• Divide both parts of the fraction by their GCD to obtain the simplest form.

For instance, in the equation \( \frac{10-2}{10-5} \), we first calculate it to get \( \frac{8}{5} \). Since 8 and 5 have no common divisors other than 1, \( \frac{8}{5} \) is already in its simplest form. Simplification is important to make fractions easier to interpret and use in calculations.