Problem 44 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$g(3)$$

Step-by-Step Solution

Answer

The value of g(3) is \(32\).

Step 1: Write the given functions

The given functions are: \(f(x) = -5x + 2\) and \(g(x) = x^2 + 7x + 2\) We are asked to find g(3).

Step 2: Substitute the value into the function g(x)

To find g(3), we need to substitute x = 3 in the expression for g(x). \(g(3) = (3)^2 + 7(3) + 2\)

Step 3: Simplify the expression

Now, let's simplify the expression and compute the value for g(3). \(g(3) = 9 + 21 + 2\)

Step 4: Calculate the result

Adding all the terms, we get: \(g(3) = 32\) So, the value of g(3) is 32.

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients. They are made up of terms, each powered by whole numbers, known as exponents. For instance, in the polynomial function \(g(x) = x^2 + 7x + 2\), the terms include \(x^2\), \(7x\), and \(2\). Each term has its own coefficient and exponent, and together they form a polynomial.

These functions are key in algebra because they can represent a vast range of phenomena in both theoretical and real-world contexts. Understanding polynomial functions is essential for solving various algebraic equations and modeling data. When working with polynomial functions, the degree of the polynomial is indicated by the highest exponent, which in our example is 2, making \(g(x)\) a quadratic polynomial.

The Substitution Method

The substitution method is a crucial technique used to evaluate functions. It involves replacing a variable with a specific value to find the function's output at that point. To apply this method, you identify the function’s expression, select the value to substitute, and calculate.

In the example \(g(x) = x^2 + 7x + 2\), we apply the substitution method by plugging in \(x = 3\) into the function. Therefore, \(g(3)\) becomes \((3)^2 + 7(3) + 2\). This allows us to compute the function's value precisely at \(x=3\).
Substitution is a foundational skill in algebra that is widely applicable, enabling students to solve equations and understand function behavior at specific points. It can simplify complex functions into numerical values that are easier to interpret.

Simplifying Algebraic Expressions

Algebraic simplification involves combining and reducing expressions to their simplest form. This process not only makes calculations easier but also helps in understanding the equation's structure. In algebra, simplification typically involves combining like terms and performing arithmetic operations.

For example, in the expression \(g(3) = 9 + 21 + 2\), the next step is to add all the numbers together. By calculating \(9 + 21 + 2\), you simplify the expression to a single value, which in this case is \(32\).
Simplification is crucial as it often reveals the most concise form of an expression, aiding further mathematical operations and understanding. It’s an essential tool for solving equations efficiently, ensuring that results are both accurate and easy to communicate.