Problem 35 - 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. $$ g(0) $$

Step-by-Step Solution

Answer

The value of \(g(0)\) is 0.

Step 1: Understand the Function g(x)

The function given is \(g(x) = \frac{x^2 + 2x}{x+3}\). We need to evaluate this function at \(x = 0\). This means wherever there is an \(x\) in the function, we substitute it with \(0\).

Step 2: Substitute x with 0 in g(x)

Replace all instances of \(x\) in the function \(g(x) = \frac{x^2 + 2x}{x+3}\) with \(0\). This will give us: \[g(0) = \frac{0^2 + 2(0)}{0+3}.\]

Step 3: Simplify the Expression

Calculate the expression \(0^2 + 2(0)\) which simplifies to \(0\). The denominator \(0 + 3\) simplifies to \(3\). Thus, the function simplifies to \[g(0) = \frac{0}{3}.\]

Step 4: Solve for g(0)

The expression \(\frac{0}{3}\) calculates to \(0\). Therefore, \(g(0) = 0\).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. These functions look like this:

• A constant term or just a number, like “5”.
• A linear term, like “2x”, which has the variable raised to the power of one.
• Quadratic terms, like “x^2”, where the variable is squared.

It's essential to know that every polynomial term is an expression multiplied by a variable. For instance, in a quadratic polynomial like "6x^2 + 3x + 2", "6x^2" is the quadratic term, "3x" is the linear term, and "2" is the constant. This understanding helps interpret how different values of the variable impact the polynomial's value.
In the function provided earlier, the expression inside the function, like those for functions f(x) and g(x), depends on the polynomial components of the numerator.

Decoding Rational Expressions

Rational expressions are essentially fractions where both the numerator and the denominator are polynomials. Just like the rational numbers you work with in fractions, rational expressions follow specific rules.

• The expression will be undefined if the denominator equals zero.
• Simplification can often remove problematic zeros in denominators, but it's best to check first.

Take this rational expression: \[g(x) = \frac{x^2 + 2x}{x+3}\]This expression is a rational function because it represents a ratio of two polynomials. When evaluating for specific values of \(x\), be cautious. For \(x = 0\), it's straightforward since adding zero doesn’t change computation much.
The critical takeaway is always to inspect the denominator. Ensuring it doesn't simplify to zero is mandatory, to avoid undefined function values.

The Essence of Algebraic Simplification

Simplification is at the heart of algebraic problem-solving, making expressions easier to interpret and compute. Simplification typically involves:

• Combining like terms where possible, e.g., adding terms with the same variable to the same power.
• Reducing fractions by canceling out common factors in the numerator and denominator.

Let’s illustrate with the function \(g(x) = \frac{x^2 + 2x}{x+3}\). Replace \(x\) with zero, simplifying to: \[g(0) = \frac{0^2 + 2 \times 0}{0+3} = \frac{0}{3}\]The result is zero because the numerator ends up as zero, and any number divided by another becomes zero. Simplifying expressions like these saves both effort and time in calculation. Always push towards the most straightforward form of an expression for accuracy and efficiency.
Remember that simplification isn’t just a technique; it’s a pathway to clearer, more effective problem-solving in algebra.