Problem 51 - Algebra A Combined Function

Question

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7,\) find each function value. \(g(2)\)

Step-by-Step Solution

Answer

The function value \(g(2)\) is 7.

Step 1: Substitute the input into the function

We are given the function \(g(x) = 4x^2 - 6x + 3\). To find \(g(2)\), substitute \(2\) for \(x\) in the expression.

Step 2: Calculate each term

First, calculate \(4 \times 2^2\). This results in \(4 \times 4 = 16\). Next, calculate \(-6 \times 2\), which is equal to \(-12\). The constant term is \(+3\).

Step 3: Sum the results

Add the results from each term: \(16 - 12 + 3\). This gives you \(16 - 12 = 4\), and \(4 + 3 = 7\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions composed of variables and constants, combined using algebraic operations such as addition, subtraction, multiplication, and division. These functions are central to algebra because they provide a way to describe mathematical relationships. They often include terms involving powers of variables, allowing you to model various real-world scenarios.
To fully understand algebraic functions, consider these key points:

• They can have one or more variables.
• These functions can be simple, like linear functions, or complex, like rational functions.
• Operations such as substitution can be used to evaluate them at specific points, giving us concrete numerical values.

By using algebraic functions, students are able to solve problems and understand how changing one variable impacts the whole equation.

Polynomial Functions

A polynomial function is a special kind of algebraic function. It consists of variables raised to whole-number powers and their coefficients. Polynomial functions can have one or more terms, such as a monomial, binomial, or trinomial.
Understanding polynomial functions is important because:

• They are the building blocks of many mathematical models, involving roots and factorization.
• They allow us to quickly find values through simple arithmetic operations.
• Higher degree polynomials describe complex relationships, adding depth to your math competency.

Considering the function from the exercise, \( g(x) = 4x^2 - 6x + 3 \), it's a quadratic function—a specific type of polynomial with the highest degree of 2, indicative of a parabolic shape when graphed.

Substitution Method

The substitution method is a powerful tool used in evaluating functions. It involves replacing the variable in a function with a specific numerical value. This method is particularly useful for finding the output of a function for given inputs, and it's extensively used not only in algebra but also in calculus and differential equations.
Why the substitution method is useful:

• It provides a precise numerical evaluation of a function, which is necessary for verification and analysis.
• By substituting specific values, one can determine how changes to the input impact the overall function.
• This method makes solving equations less abstract and more concrete, especially for polynomial functions.

In the context of the exercise, substituting \( x = 2 \) into \( g(x) = 4x^2 - 6x + 3 \) simplifies the function to evaluate \( g(2) \), helping students bridge the gap between theory and practice.