Problem 52 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=x^{2}-2 $$

Step-by-Step Solution

Answer

\( g(2) = 2 \) and \( g(3) = 7 \).

Step 1: Understand the Function

The function given is \( g(x) = x^2 - 2 \). This means that for any input \( x \), the output \( g(x) \) is calculated by squaring \( x \) and then subtracting 2.

Step 2: Calculate \( g(2) \)

To find \( g(2) \), substitute 2 into the function in place of \( x \):\[g(2) = 2^2 - 2 = 4 - 2 = 2\]So, \( g(2) = 2 \).

Step 3: Calculate \( g(3) \)

To find \( g(3) \), substitute 3 into the function in place of \( x \):\[g(3) = 3^2 - 2 = 9 - 2 = 7\]So, \( g(3) = 7 \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that involves a variable raised to the second power, or squared. Understanding quadratic functions is important as they often appear in various areas of mathematics and real-world applications. The standard form of a quadratic function is given by:

• \( g(x) = ax^2 + bx + c \)

Here, \( a \), \( b \), and \( c \) are constants, with \( a \) being non-zero to ensure the function is indeed quadratic. In our example, the function is \( g(x) = x^2 - 2 \), which simplifies to \( g(x) = 1x^2 + 0x - 2 \). Here, the coefficient \( a \) is 1, \( b \) is 0, and the constant \( c \) is -2.
These types of functions are characterized by their U-shaped graph, known as a parabola. The vertex of this parabola gives valuable insights into the maximum or minimum points of the function.
Overall, recognizing the structure and form of quadratic functions aids in both solving these types of problems and applying them in practical scenarios like physics or engineering.

Substitution Method

The substitution method is a fundamental technique in mathematics, particularly useful in evaluating functions. When you have a function and need to determine its value at specific points, the substitution method becomes really handy. In this context, substitution involves replacing the variable, typically \( x \), with a number to find the output of the function.
For example, when given a function like \( g(x) = x^2 - 2 \), if you need to find \( g(2) \), simply substitute 2 for \( x \) in the expression:

• \( g(2) = 2^2 - 2 \)

This will give \( 4 - 2 = 2 \). Similarly, for \( x = 3 \), substitute 3 to get:

• \( g(3) = 3^2 - 2 \)

This results in \( 9 - 2 = 7 \).
The substitution method is not limited to quadratic functions; it can be used with any function type to evaluate specific points.

Mathematical Operations

Mathematical operations are the building blocks needed to evaluate functions. They include basic arithmetic operations such as addition, subtraction, multiplication, and division. When working with functions, these operations help us understand and calculate the outputs for given inputs.
In our given function \( g(x) = x^2 - 2 \), we first perform the operation of squaring a number, which involves multiplication, and then follow with subtraction.

• To calculate \( g(2) \), we compute \( 2^2 = 4 \) and then subtract 2 to get 2.
• For \( g(3) \), we compute \( 3^2 = 9 \) and subtract 2 to arrive at 7.

Understanding how to perform these operations accurately is crucial as they are the steps leading to evaluating any function. Developing strong skills in these operations will enable you to perform more complex mathematical problems confidently.