Problem 33 - 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. $$ g(-6) $$

Step-by-Step Solution

Answer

The value of \( g(-6) \) is -8.

Step 1: Substitute x in g(x)

To find the value of the function \( g(x) \) at \( x = -6 \), we need to substitute \( x = -6 \) into the expression for \( g(x) \). Recall that \( g(x) = \frac{x^2 + 2x}{x+3} \). Thus, substituting, we get: \( g(-6) = \frac{(-6)^2 + 2(-6)}{(-6)+3} \).

Step 2: Evaluate the Numerator

Calculate the numerator by substituting \( x = -6 \): \(-6^2 = 36\) and \( 2(-6) = -12\). Therefore, the numerator becomes: \( 36 - 12 = 24 \).

Step 3: Evaluate the Denominator

Calculate the denominator by substituting \( x = -6 \): \(-6 + 3 = -3 \).

Step 4: Simplify the Fraction

Now, we have the fraction: \( \frac{24}{-3} \).Simplify this by dividing 24 by -3, which equals -8.

Key Concepts

Substitution

Substitution is a fundamental step in evaluating a function. It involves replacing the variable with a specific value. In this exercise, we are given a function \( g(x) = \frac{x^2 + 2x}{x+3} \). Our task is to find \( g(-6) \). To do this, substitute \( x = -6 \) into every instance of \( x \) in the expression. This helps us evaluate the function's value for that particular input.

• Start by replacing all occurrences of \( x \) with \( -6 \). It transforms the expression to \( g(-6) = \frac{(-6)^2 + 2(-6)}{-6 + 3} \).
• Substitution allows you to change an algebraic expression into a numerical one, which can be further simplified.

Understanding substitution is crucial in mathematics as it is widely used in solving equations and evaluating expressions.

Numerator and Denominator Simplification

Once the substitution is complete, the next step is simplifying the expression by working on the numerator and denominator individually. In our case, the fraction is \( \frac{24}{-3} \) after substitution. We start with each part separately to ensure clarity.

• The numerator is \( x^2 + 2x \), which becomes \( 36 - 12 \) or \( 24 \) after substituting \( x = -6 \).
• The denominator \( x + 3 \) simplifies to \(-6 + 3 = -3 \).

Breaking it down into these steps helps in understanding the simplification process. Each involves basic arithmetic operations, making sure the expression is easier to manage before simplifying the overall fraction.

Rational Functions

Rational functions are essentially fractions where both the numerator and the denominator are polynomials. Evaluating these functions often requires special attention on details, especially when substituting specific values.

• Our function, \( g(x) = \frac{x^2 + 2x}{x+3} \), is a typical rational function because it's a ratio of two polynomial expressions.
• It is important to ensure the denominator does not equate to zero, as this would make the function undefined at that point. In this exercise, we checked \( x = -6 \), and found the denominator to be \( -3 \), which is valid.

Understanding the nature of rational functions helps solve similar problems effectively and avoids common pitfalls like division by zero.

Simplifying Expressions

Simplifying expressions is a vital skill in math that makes working with functions manageable. It involves reducing the expression to its simplest form by executing all possible operations.

• Once we calculate \( g(-6) = \frac{24}{-3} \), simplifying becomes dividing \( 24 \) by \( -3 \), which gives us \(-8\).
• Always check simplified expressions for possible further reductions or factorizations to ensure simplicity.

Knowing how to simplify correctly not only helps in finding accurate answers but also in understanding the function's behavior and structure.