Problem 36 - Algebra A Combined Function

Question

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ f(2) $$

Step-by-Step Solution

Answer

The value of \(f(2)\) is \(\frac{9}{2}\).

Step 1: Identify the function

Recognize that you are working with the function \(f(x) = \frac{x^2 + 5}{x}\), and you need to evaluate this function at \(x = 2\).

Step 2: Substitute x with 2

Replace \(x\) in the function \(f(x) = \frac{x^2 + 5}{x}\) with 2. This gives \(f(2) = \frac{2^2 + 5}{2}\).

Step 3: Calculate the numerator

Calculate \(2^2 + 5\). This calculation is \(4 + 5 = 9\).

Step 4: Divide the numerator by the denominator

Divide the value obtained from the numerator \(9\) by the denominator \(2\), leading to \(\frac{9}{2}\).

Step 5: Consider simplifying the result

The fraction \(\frac{9}{2}\) is already in its simplest form. Therefore, the value of \(f(2)\) is \(\frac{9}{2}\).

Key Concepts

Function Evaluation

When tackling problems involving algebraic functions, the goal is to find out what a function equals when a specific value is put into it. This process is known as 'Function Evaluation.' It involves taking the equation of the function and replacing the variable with a number. Let's say we have a function defined as \( f(x) = \frac{x^2 + 5}{x} \). To evaluate this function at a specific point, such as \( x = 2 \), you need to replace every \( x \) in the equation with 2.
By substituting 2 into \( f(x) \), you form a new expression:

• First, compute the numerator: \( 2^2 + 5 \).
• Then, handle the denominator: just \( x \), which turns into 2.
• Finally, divide to find \( f(2) \), resulting in \( \frac{9}{2} \).

Evaluating functions in this manner helps you understand and predict the behavior of the function for given values.

Substitution Method

The Substitution Method is a straightforward yet powerful tool in algebra. It involves substituting a specific value in place of the variable in an equation or function. This technique is particularly useful when you need to find the value of a function for particular variables. Here’s a handy step-by-step guide to using the substitution method:

• Identify the function you need to work with. For example, \( f(x) = \frac{x^2 + 5}{x} \).
• Determine the value of \( x \) you need to evaluate the function at, such as \( x = 2 \).
• Replace every occurrence of \( x \) in the function with the number 2.
• Calculate the resulting expression to find \( f(2) \).

Using substitution helps simplify complex mathematical problems by allowing you to focus on one value at a time, making it a vital step in evaluating functions.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. Understanding their structure is key to efficiently evaluating them. For instance, consider the function \( f(x) = \frac{x^2 + 5}{x} \). Here, \( x^2 + 5 \) is the polynomial in the numerator and \( x \) acts as the polynomial in the denominator.
Some important aspects of rational functions include:

• They are undefined where the denominator equals zero because division by zero is not permissible. For \( f(x) \), this means \( x eq 0 \).
• Rational functions can often be simplified if there are common factors in the numerator and denominator.

When evaluating rational functions, always remember to check for any restrictions on the variable values and simplify where possible. This ensures the expression is in its simplest form, providing clearer insights into the function's properties and behavior.