Problem 52 - 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. \(g(1)\)

Step-by-Step Solution

Answer

The value of \(g(1)\) is 1.

Step 1: Identify the Function

The function you need to work with is given by \(g(x) = 4x^2 - 6x + 3\). We will evaluate this function at a specific point, \(x = 1\).

Step 2: Substitute the Value

Replace \(x\) with \(1\) in the function \(g(x)\). So it becomes: \(g(1) = 4(1)^2 - 6(1) + 3\).

Step 3: Calculate the Polynomial

First, calculate \(4(1)^2 = 4\). Then compute \(-6(1) = -6\). Now, substitute these values back into the function expression: \(4 - 6 + 3\).

Step 4: Simplify the Expression

Simplify the expression \(4 - 6 + 3\) step-by-step: \(4 - 6 = -2\), then \(-2 + 3 = 1\). So, \(g(1) = 1\).

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression that consists of variables raised to various powers, combined through addition, subtraction, and multiplication. This means each term in a polynomial is a product of a constant and a variable raised to an exponent. For example, the given polynomial function in the exercise is:

• \(g(x) = 4x^2 - 6x + 3\)

Here, the exponent of \(x\) is 2 for the first term \(4x^2\) and 1 for the second term \(-6x\). Polynomials can have multiple terms, and their degree is determined by the highest power of the variable. The degree indicates the behavior of the function, especially as values grow larger. In the function \(g(x)\), the highest power is 2, meaning it is a quadratic polynomial.
Polynomial functions are foundational in algebra and calculus as they can model real-life situations and solve diverse problems. Recognizing and evaluating polynomials is a key skill for anyone learning mathematics.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations like addition, subtraction, multiplication, and division. They help us describe patterns and relationships systematically in mathematics. The exercise involves evaluating an algebraic expression:

• \(4x^2 - 6x + 3\)

In this expression:

• \(4x^2\) is a term where 4 is the coefficient and \(x^2\) is the variable's power.
• \(-6x\) is another term with -6 as the coefficient.
• 3 is a constant term, which means it does not change with the variable \(x\).

Understanding the structure of algebraic expressions prepares us to perform operations such as addition, subtraction, or substitution. It also aids in simplifying expressions to make calculations more straightforward. Recognizing different parts of an algebraic expression is crucial for problem-solving in algebra.

Substitution Method

The substitution method is a straightforward technique used to evaluate a function at a specific point. It involves replacing the variable within an expression by a given value. In this exercise, we applied the substitution method to find \(g(1)\):

• Identify your function, here \(g(x) = 4x^2 - 6x + 3\).
• Substitute \(x = 1\) into the function, making it \(g(1) = 4(1)^2 - 6(1) + 3\).
• Follow order of operations: first solve exponents and then perform multiplication and addition/subtraction.

The substitution replaces \(x\) with 1 and allows for calculation. It's an essential skill for evaluating functions because it offers direct insight into the function's behavior at specified points. Understanding substitution provides a foundation for solving equations and analyzing functions effectively in mathematics.