Problem 32 - 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. $$ g(3) $$

Step-by-Step Solution

Answer

\( g(3) = \frac{5}{2} \).

Step 1: Understand the Function

First, we need to identify what is being asked, which is to evaluate the function \( g(x) \) at a specific point, \( g(3) \). The function \( g(x) \) is given by \( g(x) = \frac{x^2 + 2x}{x+3} \).

Step 2: Substitute x with 3

Substitute \( x = 3 \) into the function \( g(x) \). This gives us: \( g(3) = \frac{3^2 + 2 \cdot 3}{3+3} \).

Step 3: Simplify the Expression

Calculate the numerator and the denominator separately. First, compute the numerator: \( 3^2 + 2 \cdot 3 = 9 + 6 = 15 \). Next, calculate the denominator: \( 3 + 3 = 6 \).

Step 4: Perform Division

Now, divide the simplified numerator by the denominator: \( g(3) = \frac{15}{6} \).

Step 5: Simplify the Result

Simplify \( \frac{15}{6} \) by finding the greatest common divisor of 15 and 6, which is 3. Divide both the numerator and denominator by 3 to get \( \frac{15}{6} = \frac{5}{2} \).

Key Concepts

Rational Functions

A rational function is a type of function formed by dividing one polynomial by another. In simpler terms, it's like having one polynomial on top of a fraction line and another polynomial below it. This makes it quite similar to fractional numbers but with polynomials. In our exercise, the function \( g(x) = \frac{x^2 + 2x}{x+3} \) is an example of a rational function. Here, the numerator is a polynomial \( x^2 + 2x \) and the denominator is a polynomial \( x+3 \).
Understanding the structure of rational functions is crucial because each part — the numerator and the denominator — plays a role in determining the values of the function. Rational functions can have values that are undefined at certain points, usually where the denominator equals zero.

Substitution Method

The substitution method allows us to find specific values of a function by replacing the variable with a given number. To find \( g(3) \), for instance, we substitute \( x \) with 3 in the function \( g(x) \). So, the expression becomes \( g(3) = \frac{3^2 + 2 \cdot 3}{3+3} \).

• This step checks what value the function takes at a specific point.
• It's a straightforward way to evaluate functions, especially when dealing with explicit expressions.
• Substitution helps us focus directly on numbers instead of the more abstract "\( x \)".

The key is making sure all parts of the function reflect the new value to accurately determine the result.

Simplification

Simplification involves breaking down expressions into their simplest form. It's much like cleaning up a messy room, making sure everything is neat and easy to understand. After substituting \( x \) with 3, we need to simplify \( \frac{3^2 + 2 \cdot 3}{3+3} \).
First, calculate the numerator: \( 3^2 + 2 \cdot 3 = 9 + 6 = 15 \). Then, calculate the denominator: \( 3 + 3 = 6 \).
Therefore, the simplified form is \( \frac{15}{6} \). Simplifying might also involve reducing fractions to their lowest terms, which we do in the next step.

Numerical Evaluation

Numerical evaluation takes us to the final part where we determine the actual numerical value of the expression. From simplification, we have \( \frac{15}{6} \).
To simplify this fraction further, find the greatest common divisor (GCD) of 15 and 6, which is 3. Divide both the numerator and the denominator by this GCD:

• Numerator: \( 15 \div 3 = 5 \)
• Denominator: \( 6 \div 3 = 2 \)

The simplified fraction is \( \frac{5}{2} \).
This means that the evaluated function \( g(3) \) is \( \frac{5}{2} \) or 2.5. Through numerical evaluation, we move from algebraic expressions to tangible numbers, making the function easier to interpret and use.